Rewrite the sum as a single logarithm. Further simplify if possible.log⁡513+log⁡512y\$$log_5$$\frac13+\log_512y

log⁡5(13+12y)\$$log_5$$\left(\frac13+12y\right)

Use the power property to rewrite the log expression.log⁡2(x+1)2\$$log_2$$\left(x+1\$$right)^2$$

2log⁡2(x+1)2\$$log_2$$\left(x+1\right)

(x+1)log⁡22\left(x+1\right)\$$log_22$$

2+log⁡2(x+1)2+\$$log_2$$\left(x+1\right)

Rewrite the log expression as the sum or difference of multiple logs.log⁡10(8x35y)\log_{10}\left(\frac{$$8x^3$$}{5y}\right)

3log⁡102+3log⁡10x−log⁡105−log⁡10y3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}5$$-\$$log_{10}y $$

3log⁡102+3log⁡10x−log⁡105+log⁡10y3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}5$$+\$$log_{10}y $$

−3log⁡102+3log⁡10x−log⁡10y-3\$$log_{10}2+3$$\$$log_{10}x$$-\$$log_{10}y $$

−3log⁡102+3log⁡10x+log⁡10y-3\$$log_{10}2+3$$\$$log_{10}x$$+\$$log_{10}y $$

Rewrite the log expression as the sum or difference of multiple logs.log⁡2(3a2b45c4)\$$log_2$$\left(\frac{$$3a^2$b^4$$}{\sqrt{$$5c^4$$}}\right)

log⁡23+2log⁡2a+4log⁡2b−12log⁡25−2log⁡2c\$$log_23+2$$\log_2a+4\log_2b-\frac12\$$log_25-2$$\log_2c

log⁡23+2log⁡2a+4log⁡2b−12log⁡25+4log⁡2c\$$log_23+2$$\log_2a+4\log_2b-\frac12\$$log_25+4$$\log_2c

log⁡23+2log⁡2a+4log⁡2b−2log⁡25−12log⁡2c\$$log_23+2$$\log_2a+4\log_2b-2\$$log_25$$-\frac12\log_2c

log⁡23+2log⁡2a+4log⁡2b−2log⁡25+12log⁡2c\$$log_23+2$$\log_2a+4\log_2b-2\$$log_25$$+\frac12\log_2c

Rewrite the log expression as a single log.3log⁡4a−12log⁡4(b+1)+log⁡4(4c)3\log_4a-\frac12\$$log_4$$\left(b+1)+\$$log_4$$\left(4c\right)\right.

log⁡4(4a3cb+1)\$$log_4$$\left(\frac{$$4a^3c$$}{\sqrt{b+1}}\right)

log⁡4(12acb+1)\$$log_4$$\left(\frac{12ac}{\sqrt{b+1}}\right)

log⁡4(4a3cb+1)\$$log_4$$\left(\frac{$$4a^3c$$}{\sqrt{b}+1}\right)

log⁡4(12acb+1)\$$log_4$$\left(\frac{12a^{}c}{\sqrt{b}+1}\right)

13. Inverse, Exponential, & Logarithmic Functions

Properties of Logarithms

13. Inverse, Exponential, & Logarithmic Functions

Properties of Logarithms: Videos & Practice Problems

Video Lessons Worksheet

Topic summary

Properties of Logarithms focus on rewriting logarithmic expressions by expanding a single log or condensing multiple logs. The central rules are the product property, quotient property, and power property: $ \log_b(mn)=\log_b(m)+\log_b(n) $, $ \log_b\\!\left(\frac{m}{n}\right)=\log_b(m)-\log_b(n) $, and $ \log_b(m^n)=n\log_b(m) $.

These properties work in both directions, but when condensing, apply the power property first, then combine logs with the same base using product or quotient rules. Roots and fractions can be rewritten with rational or negative exponents before using the power rule. A key idea is that these properties apply to multiplication, division, and powers inside the argument, not to addition or subtraction inside the argument. Expressions like $\log_b(m+n)$ or $\log_b(m-n)$ cannot be split using log properties.

Concept

Product Property of Logs

Video duration:

Product Property of Logs Video Summary

Logarithms are exponents, and their properties stem from the corresponding rules of exponents. One fundamental property is the product property of logarithms, which states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors. Specifically, for any positive numbers m and n and base b (where b ≠ 1), the product property is written as:

\[\log_b(m \times n) = \log_b m + \log_b n\]

This property mirrors the exponent rule where multiplying exponential expressions with the same base results in adding their exponents. For example, applying this property to \[\log_2(4 \times 6)\] allows us to rewrite it as \[\log_2 4 + \log_2 6\], converting multiplication inside the logarithm into addition outside.

The product property can be used in two directions: to expand a single logarithm with a product inside into a sum of logarithms, or to condense a sum of logarithms with the same base into a single logarithm of a product. It is crucial that the logarithms involved share the same base to apply this property correctly. For instance, \[\log_5(3x)\] can be expanded as \[\log_5 3 + \log_5 x\], while \[\log_{10} 7 + \log_{10} 9\] can be condensed into \[\log_{10} (7 \times 9) = \log_{10} 63\].

However, if the logarithms have different bases, such as \[\log_2 x + \log_3 8\], the product property does not apply, and no simplification using this property is possible.

It is also important to avoid the common mistake of assuming that the logarithm of a sum equals the sum of logarithms. In other words, expressions like \[\log_b (m + n)\] cannot be simplified to \[\log_b m + \log_b n\]. For example, \[\log_{10} (7 + 9)\] is not equal to \[\log_{10} 7 + \log_{10} 9\]; instead, the latter equals \[\log_{10} (7 \times 9)\] due to the product property.

Understanding and correctly applying the product property of logarithms is essential for simplifying logarithmic expressions and solving logarithmic equations efficiently.

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Example

Product Property of Logs Example 1

Video duration:

Product Property of Logs Example 1 Video Summary

When simplifying logarithmic expressions such as log base 10 of 2 plus log base 10 of 5, the product property of logarithms is essential. This property states that the sum of two logarithms with the same base can be combined into a single logarithm by multiplying their arguments. Mathematically, this is expressed as:

\[\log_{10} 2 + \log_{10} 5 = \log_{10} (2 \times 5)\]

Multiplying the arguments inside the logarithm gives:

\[\log_{10} 10\]

Recognizing that the base of the logarithm and its argument are identical is crucial. The logarithmic identity states that for any base b (where b > 0 and b ≠ 1),

\[\log_b b = 1\]

Applying this property simplifies the expression to:

\[\log_{10} 10 = 1\]

This process highlights the importance of understanding and applying logarithmic properties such as the product rule and the identity involving equal base and argument. Mastery of these concepts allows for efficient simplification of logarithmic expressions, which is fundamental in algebra and higher-level mathematics.

Problem

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$l o g sub 10 of open paren 5 times 7 close paren$

$\log_{10} 5 \times \log_{10} 7$

$\log_{10} 5 + \log_{10} 7 + \log_{10} 35$

$\log_{10} 5 + \log_{10} 7$

$\log_{10} 7 - \log_{10} 5$

Problem

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$the log base 4 of 4 x$

$\log_{4} 1 + \log_{4} x$

$1 + \log_{4} x$

$\log_{4} 1 + \log_{4} x + 1$

$\log_{4} 1 \times \log_{4} x$

Problem

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$the log base 3 of 10 m n$

$\log_{3} 10 \times \log_{3} m \times \log_{3} n$

$the log base 10 of m n$

$\log_{3} 10 + \log_{3} m + \log_{3} n$

$\log_{10} m + \log_{10} n + \log_{10} 3$

Problem

Rewrite the sum as a single logarithm. Further simplify if possible.
$\log_{3} 4 + \log_{3} 9$

$the log base 3 of 36$

$the log base 3 of 13$

$the log base 3 of 1$

$the log base 10 of 36$

Problem

Rewrite the sum as a single logarithm. Further simplify if possible.
$\log_{5} \frac{1}{3} + \log_{5} 12 y$

$\log_{5} (\frac{1}{3} + 12 y)$

$the log base 5 of 4 y$

$the log base 5 of 36 y$

$the log base 5 of 4$

Problem

Rewrite the sum as a single logarithm. Further simplify if possible.
$\log_{10} x + \log_{10} (x + 2)$

$the log base 10 of 2 x cubed$

$the log base 10 of 4 x$

$\log_{10} (x^{2} + 2 x)$

$\log_{10} (x^{2} + 2)$

Concept

Quotient Property of Logs

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Quotient Property of Logs Video Summary

The quotient property of logarithms is a fundamental rule that allows us to simplify the logarithm of a quotient by expressing it as the difference of two logarithms with the same base. This property is directly derived from the quotient rule of exponents, which states that when dividing exponential expressions with the same base, you subtract the exponents. Similarly, for logarithms, if you have log base b of a quotient, such as $\log_b \left(\frac{m}{n}\right)$, it can be rewritten as the difference of two logarithms: $\log_b m - \log_b n$. This transformation turns division inside the logarithm into subtraction outside the logarithm.

For example, $\log_3 \left(\frac{5}{6}\right)$ can be expanded using the quotient property to $\log_3 5 - \log_3 6$. This property works both ways: you can expand a single logarithm of a quotient into a difference of logarithms, or you can condense a difference of logarithms with the same base into a single logarithm of a quotient.

When applying the quotient property, it is essential that the logarithms involved share the same base. For instance, $\log_5 32 - \log_5 8$ can be combined into $\log_5 \left(\frac{32}{8}\right)$, which simplifies further to $\log_5 4$. Similarly, expressions with variables, such as $\log_2 x - \log_2 (x + 4)$, can be rewritten as $\log_2 \left(\frac{x}{x + 4}\right)$ by applying the quotient property.

It is important to avoid a common misconception: the quotient property does not apply to subtraction inside the argument of a logarithm. Specifically, $\log_b (m - n)$ cannot be separated into $\log_b m - \log_b n$. The quotient property only applies when the argument involves division, not subtraction. Therefore, expressions with subtraction inside the logarithm must be left as they are, as there is no logarithmic property to simplify them further.

Understanding and correctly applying the quotient property of logarithms enhances the ability to simplify complex logarithmic expressions, solve logarithmic equations, and deepen comprehension of the relationship between exponents and logarithms.

Problem

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.
$\log_{2} (\frac{x}{6})$

$\log_{2} 6 - \log_{2} x$

$\frac{\log_{2} x}{\log_{2} 6}$

$\frac{\log_{2} 6}{\log_{2} x}$

$\log_{2} x - \log_{2} 6$

Problem

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.
$log base 12 one fourth$

$- \log_{12} 3$

$the log base 12 of 3$

$the log base 12 of 4$

$- \log_{12} 4$

Problem

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.
$\log_{5} (\frac{125}{x})$

$3 - \log_{5} x$

$\frac{\log_{5} 125}{\log_{5} x}$

$\log_{5} x - 3$

$\frac{\log_{5} x}{\log_{5} 125}$

Problem

Rewrite the difference as a single logarithm. Further simplify if possible.
$\log_{5} 24 - \log_{5} 8$

$the log base 5 of 3$

$log base 5 one third$

$the log base 5 of 16$

$- \log_{5} 16$

Problem

Rewrite the difference as a single logarithm. Further simplify if possible.
$\log_{10} (8 x^{2}) - \log_{10} (2 x)$

$the log base 10 of 4 x$

$the log base 10 of 4 x squared$

$the log base 10 of 32 x$

$\log_{10} (8 x^{2} - 2 x)$

Concept

Power Property of Logs

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Power Property of Logs Video Summary

The power property of logarithms is a fundamental rule that allows you to simplify expressions where a logarithm contains an argument raised to a power. Specifically, if you have a logarithm of the form $\log_b (m^n)$, you can rewrite it by bringing the exponent $n$ in front of the logarithm as a multiplier, resulting in $n \times \log_b (m)$. This property is especially useful for breaking down complex logarithmic expressions into simpler terms.

For example, consider $\log_3 (7^2)$. Using the power property, this can be rewritten as $2 \times \log_3 (7)$, effectively pulling the exponent 2 out front. This transformation helps in both simplifying calculations and solving logarithmic equations.

Importantly, the power property works in both directions. If you see a logarithm multiplied by a constant, such as $4 \times \log_5 (x)$, you can rewrite it by moving the multiplier inside the logarithm as an exponent: $\log_5 (x^4)$. This flexibility allows for easier manipulation of logarithmic expressions depending on the problem context.

Applying exponent rules alongside the power property enhances your ability to simplify logarithms. For instance, the square root of a number can be expressed as an exponent of one-half, so $\log_2 (\sqrt{5})$ becomes $\log_2 (5^{1/2})$, which then simplifies to $\frac{1}{2} \times \log_2 (5)$ using the power property.

Similarly, when dealing with fractions inside logarithms, negative exponents come into play. For example, $\log_{10} \left(\frac{1}{x^3}\right)$ can be rewritten as $\log_{10} (x^{-3})$, which simplifies to $-3 \times \log_{10} (x)$ by applying the power property.

Understanding and applying the power property of logarithms, along with exponent rules, is essential for mastering logarithmic expressions. This property not only simplifies calculations but also lays the groundwork for solving more complex logarithmic equations and combining multiple logarithmic properties effectively.

Example

Power Property of Logs Example 2

Video duration:

48s

Power Property of Logs Example 2 Video Summary

When simplifying logarithmic expressions involving roots and exponents, it is helpful to rewrite roots as rational exponents. For example, the cube root of y raised to the fourth power can be expressed as y raised to the power of ⁴⁄₃. This is because the cube root corresponds to an exponent of ¹⁄₃, so applying it to y⁴ results in y^4·1/3 = y^4/3.

Using the power property of logarithms, which states that log_b(aⁿ) = n × log_b(a), the expression log₁₀(y^4/3) can be rewritten as ⁴⁄₃ × log₁₀(y). This property allows the exponent to be brought in front of the logarithm as a multiplier, simplifying the expression and making it easier to work with.

In summary, converting roots to rational exponents and applying the power property of logarithms are key strategies for rewriting and simplifying logarithmic expressions. This approach enhances understanding and manipulation of logarithmic functions, especially when dealing with complex exponents.

Problem

Use the power property to rewrite the log expression.
$\log_{9} 5^{4}$

$5 the log base 9 of 4$

$5 the log base 4 of 9$

$4 the log base 9 of 5$

$4 the log base 5 of 9$

Problem

Use the power property to rewrite the log expression.
$\log_{6} \frac{1}{\sqrt{m}}$

$one half the log base 6 of m$

$2 the log base 6 of m$

$- 2 \log_{6} m$

$- \frac{1}{2} \log_{6} m$

Problem

Use the power property to rewrite the log expression.
$\log_{2} {(x + 1)}^{2}$

$x plus 1$

$2 \log_{2} (x + 1)$

$(x + 1) \log_{2} 2$

$2 + \log_{2} (x + 1)$

Concept

Expanding & Condensing Log Expressions Using Multiple Properties

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Expanding & Condensing Log Expressions Using Multiple Properties Video Summary

Logarithmic expressions can be expanded or condensed by applying fundamental log properties such as the product, quotient, and power rules. When expanding a logarithmic expression, the goal is to rewrite it as the sum or difference of multiple logarithms. For example, consider the expression log base 10 of 3xy². Using the product property, this can be separated into the sum of logs: $ \log_{10}(3) + \log_{10}(x) + \log_{10}(y^2) $. Then, applying the power property allows the exponent to be brought in front as a multiplier, resulting in $ \log_{10}(3) + \log_{10}(x) + 2 \log_{10}(y) $. This fully expanded form breaks down the original logarithm into simpler components.

Conversely, condensing logarithmic expressions involves combining multiple logs into a single logarithm. This process typically starts with the power property, which converts coefficients in front of logs into exponents within the argument. For instance, given the expression $ 4 \log_5(x) - \log_5(x + 2) $, applying the power rule first transforms it into $ \log_5(x^4) - \log_5(x + 2) $. Then, the quotient property combines the subtraction of logs into the logarithm of a quotient: $ \log_5 \left( \frac{x^4}{x + 2} \right) $. This condensed form simplifies the expression into a single logarithm.

Understanding and applying these logarithmic properties—product, quotient, and power—enables the manipulation of complex logarithmic expressions for simplification or expansion. The product property states that the logarithm of a product equals the sum of the logarithms: $ \log_b(MN) = \log_b(M) + \log_b(N) $. The quotient property expresses the logarithm of a quotient as the difference of logarithms: $ \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) $. The power property allows exponents inside the logarithm to be moved as coefficients: $ \log_b(M^p) = p \log_b(M) $. Mastery of these properties is essential for solving logarithmic equations, simplifying expressions, and understanding the behavior of logarithmic functions.

Problem

Rewrite the log expression as the sum or difference of multiple logs.
$\log_{10} (\frac{8 x^{3}}{5 y})$

$3 \log_{10} 2 + 3 \log_{10} x - \log_{10} 5 + \log_{10} y$

$- 3 \log_{10} 2 + 3 \log_{10} x - \log_{10} y$

$3 \log_{10} 2 + 3 \log_{10} x - \log_{10} 5 - \log_{10} y$

$- 3 \log_{10} 2 + 3 \log_{10} x + \log_{10} y$

Problem

Rewrite the log expression as the sum or difference of multiple logs.
$\log_{2} (\frac{3 a^{2} b^{4}}{\sqrt{5 c^{4}}})$

$\log_{2} 3 + 2 \log_{2} a + 4 \log_{2} b - \frac{1}{2} \log_{2} 5 - 2 \log_{2} c$

$\log_{2} 3 + 2 \log_{2} a + 4 \log_{2} b - \frac{1}{2} \log_{2} 5 + 4 \log_{2} c$

$\log_{2} 3 + 2 \log_{2} a + 4 \log_{2} b - 2 \log_{2} 5 - \frac{1}{2} \log_{2} c$

$\log_{2} 3 + 2 \log_{2} a + 4 \log_{2} b - 2 \log_{2} 5 + \frac{1}{2} \log_{2} c$

Problem

Rewrite the log expression as a single log.
$\log_{3} 5 + 2 \log_{3} x - \log_{3} 2$

$\log_{3} \frac{5 - 2 x}{2}$

$the log base 3 of 5 x$

$\log_{3} \frac{5 x^{2}}{2}$

$\log_{3} \frac{5 + 2 x}{2}$

Problem

Rewrite the log expression as a single log.
$3 \log_{4} a - \frac{1}{2} \log_{4} (b + 1) + \log_{4} (4 c)$

$\log_{4} (\frac{12 a c}{\sqrt{b + 1}})$

$\log_{4} (\frac{4 a^{3} c}{\sqrt{b + 1}})$

$\log_{4} (\frac{4 a^{3} c}{\sqrt{b} + 1})$

$\log_{4} (\frac{12 a^{} c}{\sqrt{b} + 1})$

Example

Expanding & Condensing Log Expressions Using Multiple Properties Example 3

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Expanding & Condensing Log Expressions Using Multiple Properties Example 3 Video Summary

When working with logarithmic expressions involving an unknown base, it is essential to use logarithm properties to rewrite complex expressions in terms of known logarithms. Given values such as $\log_x 3 = -1.585$ and $\log_x 5 = -2.322$, you can approximate more complicated logarithms by expressing their arguments as products, quotients, or powers of 3 and 5.

For example, to approximate $\log_x \frac{45}{\sqrt{3}}$, start by applying the quotient property of logarithms:

\[\log_x \frac{45}{\sqrt{3}} = \log_x 45 - \log_x \sqrt{3}\]

Next, factor 45 into prime factors related to 3 and 5:

\[45 = 5 \times 3^2\]

Using the product property, this becomes:

\[\log_x 45 = \log_x 5 + \log_x 3^2\]

For the square root, rewrite it as a fractional exponent and apply the power property:

\[\log_x \sqrt{3} = \log_x 3^{\frac{1}{2}} = \frac{1}{2} \log_x 3\]

Putting it all together:

\[\log_x \frac{45}{\sqrt{3}} = \log_x 5 + 2 \log_x 3 - \frac{1}{2} \log_x 3 = \log_x 5 + \frac{3}{2} \log_x 3\]

Substitute the given values:

\[= -2.322 + \frac{3}{2} \times (-1.585) = -2.322 - 2.378 = -4.700\]

Rounded to three decimal places, the approximation is -4.700.

Similarly, to approximate $\log_x \frac{75}{3 \sqrt{5}}$, begin by applying the quotient property:

\[\log_x \frac{75}{3 \sqrt{5}} = \log_x 75 - \log_x (3 \sqrt{5})\]

Express 75 in terms of 3 and 5:

\[75 = 3 \times 5^2\]

Apply the product property to the denominator:

\[\log_x (3 \sqrt{5}) = \log_x 3 + \log_x 5^{\frac{1}{2}} = \log_x 3 + \frac{1}{2} \log_x 5\]

Rewrite the numerator:

\[\log_x 75 = \log_x 3 + \log_x 5^2 = \log_x 3 + 2 \log_x 5\]

Substitute these back into the expression:

\[\log_x \frac{75}{3 \sqrt{5}} = (\log_x 3 + 2 \log_x 5) - (\log_x 3 + \frac{1}{2} \log_x 5) = 2 \log_x 5 - \frac{1}{2} \log_x 5 = \frac{3}{2} \log_x 5\]

Plugging in the known value:

\[\frac{3}{2} \times (-2.322) = -3.483\]

Rounded to three decimal places, the approximation is -3.483.

These examples demonstrate how logarithmic properties—product, quotient, and power rules—allow you to simplify complex logarithmic expressions into sums and differences of known logarithms. This approach is especially useful when the base is unknown but certain logarithmic values are provided. By breaking down arguments into prime factors and applying these properties, you can efficiently approximate logarithmic values to the desired precision.

Problem

Determine if the given log statement is true or false.
$\log_{2} 48 = \log_{2} 3 + 4 \log_{2} 2$

True

False

Cannot be determined

Problem

Determine if the given log statement is true or false.
$\log_{3} (\frac{324}{4 \sqrt{3}}) = 4 \log_{3} 3 + \log_{3} 3 - \frac{1}{2} \log_{3} 4$

True

False

Cannot be determined

Here's what students ask on this topic:

The product property of logarithms states that $\log_{b} (m \times n) = \log_{b} (m) + \log_{b} (n)$ . This means that the logarithm of a product can be rewritten as the sum of the logarithms of the factors, provided both logs have the same base $b$ . When expanding, you take a single log with a product inside and write it as the sum of multiple logs. Conversely, when condensing, you combine the sum of logs into one log with the product of their arguments. For example, $\log_{2} (4×6) = \log_{2} (4) + \log_{2} (6)$ . This property is useful for simplifying logarithmic expressions and solving logarithmic equations.

The quotient property of logarithms states that $\log_{b} (\frac{m}{n}) = \log_{b} (m) - \log_{b} (n)$ . This means the logarithm of a quotient can be rewritten as the difference of the logarithms of numerator and denominator, assuming the logs share the same base $b$ . You can apply this property when you have division inside the argument of a single log or when condensing the difference of two logs into one log with a quotient inside. For example, $\log_{3} (\frac{5}{6}) = \log_{3} (5) − \log_{3} (6)$ . Note that subtraction inside the argument (like $\log_{b} (m−n)$ ) cannot be simplified using this property.

The power property of logarithms states that $\log_{b} (m^{n}) = n \log_{b} (m)$ . This means that when the argument of a logarithm is raised to a power, you can pull the exponent out in front of the log as a multiplier. This property works both ways: you can expand a log with a power inside by moving the exponent out front, or condense a coefficient multiplying a log by moving it as an exponent inside the argument. For example, $\log_{3} (7)^{2} = 2 × \log_{3} (7)$ . This property is especially useful when dealing with roots or fractional exponents, which can be rewritten as powers before applying the property.

To handle complex logarithmic expressions, you often need to apply multiple properties—product, quotient, and power—in sequence. When expanding, start by using the product and quotient properties to break down multiplication and division inside the log into sums and differences of logs. Then apply the power property to move exponents out front as multipliers. For example, $\log_{10} (3×x×y)^{2})$ expands to $\log_{10} (3) + \log_{10} (x) + 2 × \log_{10} (y)$ . When condensing, apply the power property first by moving coefficients inside as exponents, then use product and quotient properties to combine sums and differences into single logs with multiplication or division inside. This systematic approach ensures accurate simplification of logarithmic expressions.

Logarithmic properties like the product, quotient, and power rules only apply to multiplication, division, and powers inside the argument of a logarithm. They do not apply to addition or subtraction inside the argument because logarithms are defined based on exponents, and exponent rules do not translate addition or subtraction inside the argument into sums or differences of logs. For example, $\log_{10} (7 + 9)$ cannot be rewritten as $\log_{10} (7) + \log_{10} (9)$ . Attempting to do so is a common mistake. The properties only convert multiplication to addition and division to subtraction, reflecting the underlying exponent rules. Therefore, addition or subtraction inside the argument must be handled differently, often by simplifying the expression inside the log before applying logarithmic properties.

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Properties of Logarithms: Videos & Practice Problems

Product Property of Logs

Product Property of Logs Video Summary

Product Property of Logs Example 1

Product Property of Logs Example 1 Video Summary

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.log10(5⋅7)log_{10}\(\left\)(5\(\cdot\)7\(\right\))

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.log44x\(\log\)_44x

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.log310mn\(\log\)_310mn

Rewrite the sum as a single logarithm. Further simplify if possible.log34+log39\(\log\)_34+\(\log\)_39

Rewrite the sum as a single logarithm. Further simplify if possible.log513+log512y\(\log\)_5\(\frac\)13+\(\log\)_512y

Rewrite the sum as a single logarithm. Further simplify if possible.log10x+log10(x+2)\(\log\)_{10}x+\(\log\)_{10}\(\left\)(x+2)\(\right\).

Quotient Property of Logs

Quotient Property of Logs Video Summary

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.log2(x6)\(\log\)_2\(\left\)(\(\frac{x}{6}\]\right\))

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.log12(14)\(\log\)_{12}\(\left\)(\(\frac\)14\(\right\))

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.log5(125x)\(\log\)_5\(\left\)(\(\frac{125}{x}\]\right\))

Rewrite the difference as a single logarithm. Further simplify if possible.log524−log58\(\log\)_524-\(\log\)_58

Rewrite the difference as a single logarithm. Further simplify if possible.log10(8x2)−log10(2x)\(\log\)_{10}\(\left\)(8x^2)-\(\log\)_{10}\(\left\)(2x)\(\right\).\(\right\).

Power Property of Logs

Power Property of Logs Video Summary

Power Property of Logs Example 2

Power Property of Logs Example 2 Video Summary

Use the power property to rewrite the log expression.log954\(\log\)_95^4

Use the power property to rewrite the log expression.log61m\(\log\)_6\(\frac{1}{\sqrt{m}\)}

Use the power property to rewrite the log expression.log2(x+1)2\(\log\)_2\(\left\)(x+1\(\right\))^2

Expanding & Condensing Log Expressions Using Multiple Properties

Expanding & Condensing Log Expressions Using Multiple Properties Video Summary

Rewrite the log expression as the sum or difference of multiple logs.log10(8x35y)\(\log\)_{10}\(\left\)(\(\frac{8x^3}{5y}\]\right\))

Rewrite the log expression as the sum or difference of multiple logs.log2(3a2b45c4)\(\log\)_2\(\left\)(\(\frac{3a^2b^4}{\sqrt{5c^4}\)}\(\right\))

Rewrite the log expression as a single log.log35+2log3x−log32\(\log\)_35+2\(\log\)_3x-\(\log\)_32

Rewrite the log expression as a single log.3log4a−12log4(b+1)+log4(4c)3\(\log\)_4a-\(\frac\)12\(\log\)_4\(\left\)(b+1)+\(\log\)_4\(\left\)(4c\(\right\))\(\right\).

Expanding & Condensing Log Expressions Using Multiple Properties Example 3

Expanding & Condensing Log Expressions Using Multiple Properties Example 3 Video Summary

Determine if the given log statement is true or false. log248=log23+4log22\(\log\)_248=\(\log\)_23+4\(\log\)_22

Determine if the given log statement is true or false. log3(32443)=4log33+log33−12log34\(\log\)_3\(\left\)(\(\frac{324}{4\sqrt3}\]\right\))=4\(\log\)_33+\(\log\)_33-\(\frac\)12\(\log\)_34

Here's what students ask on this topic:

What is the product property of logarithms and how is it used to expand or condense logarithmic expressions?

What is the product property of logarithms and how is it used to expand or condense logarithmic expressions?

How does the quotient property of logarithms work and when can it be applied?

How does the quotient property of logarithms work and when can it be applied?

What is the power property of logarithms and how can it be applied to simplify logarithmic expressions?

What is the power property of logarithms and how can it be applied to simplify logarithmic expressions?

How do you combine multiple logarithmic properties to expand or condense complex logarithmic expressions?

How do you combine multiple logarithmic properties to expand or condense complex logarithmic expressions?

Why can't logarithmic properties be applied to addition or subtraction inside the argument of a logarithm?

Why can't logarithmic properties be applied to addition or subtraction inside the argument of a logarithm?

Your Beginning & Intermediate Algebra tutors

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$l o g sub 10 of open paren 5 times 7 close paren$

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$the log base 4 of 4 x$

Rewrite the log expression as a sum of multiple logs. Further simplify if possible.
$the log base 3 of 10 m n$

Rewrite the sum as a single logarithm. Further simplify if possible.
$\log_{3} 4 + \log_{3} 9$

Rewrite the sum as a single logarithm. Further simplify if possible.
$\log_{5} \frac{1}{3} + \log_{5} 12 y$

Rewrite the sum as a single logarithm. Further simplify if possible.
$\log_{10} x + \log_{10} (x + 2)$

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.
$\log_{2} (\frac{x}{6})$

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.
$log base 12 one fourth$

Rewrite the log expression as a difference of multiple logs. Further simplify if possible.
$\log_{5} (\frac{125}{x})$

Rewrite the difference as a single logarithm. Further simplify if possible.
$\log_{5} 24 - \log_{5} 8$

Rewrite the difference as a single logarithm. Further simplify if possible.
$\log_{10} (8 x^{2}) - \log_{10} (2 x)$

Use the power property to rewrite the log expression.
$\log_{9} 5^{4}$

Use the power property to rewrite the log expression.
$\log_{6} \frac{1}{\sqrt{m}}$

Use the power property to rewrite the log expression.
$\log_{2} {(x + 1)}^{2}$

Rewrite the log expression as the sum or difference of multiple logs.
$\log_{10} (\frac{8 x^{3}}{5 y})$

Rewrite the log expression as the sum or difference of multiple logs.
$\log_{2} (\frac{3 a^{2} b^{4}}{\sqrt{5 c^{4}}})$

Rewrite the log expression as a single log.
$\log_{3} 5 + 2 \log_{3} x - \log_{3} 2$

Rewrite the log expression as a single log.
$3 \log_{4} a - \frac{1}{2} \log_{4} (b + 1) + \log_{4} (4 c)$

Determine if the given log statement is true or false.
$\log_{2} 48 = \log_{2} 3 + 4 \log_{2} 2$

Determine if the given log statement is true or false.
$\log_{3} (\frac{324}{4 \sqrt{3}}) = 4 \log_{3} 3 + \log_{3} 3 - \frac{1}{2} \log_{3} 4$