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⁡61m\$$log_6$$\frac{1}{\sqrt{m}}

−12log⁡6m-\frac12\log_6m−21log6m

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)2log2(x+1)

(x+1)log⁡22\left(x+1\right)\$$log_22(x+1)log2$$2

2+log⁡2(x+1)2+\$$log_2$$\left(x+1\right)2+log2(x+1)

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$$-\$$log_{10}y3log10$$2+3log10x−log105−log10y

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$$−3log102+3log10x+log10y

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_2clog23+2log2a+4log2b−21log25−2log2c

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)log4(b+14a3c)

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)log4(b+14a3c)

log⁡4(12acb+1)\$$log_4$$\left(\frac{12a^{}c}{\sqrt{b}+1}\right)log4(b+112ac)

11. Inverse, Exponential, & Logarithmic Functions

Properties of Logarithms

11. Inverse, Exponential, & Logarithmic Functions

Properties of Logarithms: Videos & Practice Problems

Video Lessons Worksheet

Topic summary

Properties of Logarithms focus on three main rules used to expand or condense logarithmic expressions: the product property, quotient property, and power property. These rules connect logarithms to exponent behavior. The product property says $ \log_b(mn)=\log_b(m)+\log_b(n) $ , and the quotient property says $ \log_b\\!\left(\frac{m}{n}\right)=\log_b(m)-\log_b(n) $ .

The power property states $ \log_b(m^n)=n\log_b(m) $ . It can also be used in reverse when condensing, and it is usually applied first. Exponent rules are important, so roots can be rewritten as rational exponents and factors in a denominator can be written with negative exponents. These properties only apply to multiplication, division, or powers in the argument, not to addition or subtraction inside the argument. When condensing multiple logs, the logs must have the same base.

Concept

Product Property of Logs

Video duration:

Product Property of Logs Video Summary

Logarithms are exponents, and their properties closely mirror those 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 given by:

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

This property arises because multiplying exponential expressions with the same base corresponds to adding their exponents. For example, applying this to logarithms, the multiplication inside the logarithm’s argument translates to addition outside the logarithm.

Consider the example: \[\log_2(4 \times 6)\] Using the product property, this can be rewritten as:

\[\log_2 4 + \log_2 6\]

This shows how multiplication inside a single logarithm can be expanded into the sum of two logarithms with the same base.

Importantly, the product property works both ways. It can be used to expand a single logarithm into multiple logarithms or to condense multiple logarithms with the same base into a single logarithm. For instance, if you have:

\[\log_5 3 + \log_5 x\]

you can condense this into:

\[\log_5 (3x)\]

However, this reverse application requires that all logarithms involved share the same base. If the bases differ, such as in:

\[\log_2 x + \log_3 8\]

the product property cannot be applied, and the expression cannot be simplified further using this property.

It is also crucial to avoid a common misconception: the logarithm of a sum is not equal to the sum of the logarithms. In other words,

\[\log_b (m + n) \neq \log_b m + \log_b n\]

This means expressions like \[\log_{10} (7 + 9)\] cannot be rewritten as \[\log_{10} 7 + \log_{10} 9\]. Instead, the product property applies only when the operation inside the logarithm is multiplication, not addition.

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 whose argument is the product of the original 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 the same 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:

\[1\]

This process highlights the importance of understanding and applying logarithmic properties such as the product rule and the fundamental identity where the logarithm of a base to itself equals one. Mastery of these concepts allows for efficient simplification of logarithmic expressions and deepens comprehension of logarithmic functions.

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$_41+$\log$_4x+1$

$$\log$_41$\times\]\log$_4x$

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$_310+$\log$_3m+$\log$_3n$

$$\log$_{10}m+$\log$_{10}n+$\log$_{10}3$

Problem

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

$$\log$_336$

$$\log$_313$

$$\log$_31$

$$\log$_{10}36$

Problem

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

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

$$\log$_54y$

$$\log$_536y$

$$\log$_54$

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$_{10}$\left$(x^2+2x$\right$)$

$$\log$_{10}$\left$(x^2+2$\right$)$

Concept

Quotient Property of Logs

Video duration:

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 expressed as $\log_3 5 - \log_3 6$. This property works both ways: you can expand a single logarithm of a quotient into a subtraction of logarithms, or you can condense a subtraction 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, $\log_{10} \left(\frac{7}{x}\right)$ can be rewritten as $\log_{10} 7 - \log_{10} x$. Conversely, if you have $\log_5 32 - \log_5 8$, you can combine these into a single logarithm: $\log_5 \left(\frac{32}{8}\right)$, which simplifies further to $\log_5 4$.

Variables can also be included in the arguments of logarithms when using the quotient property. For example, $\log_2 x - \log_2 (x + 4)$ simplifies to $\log_2 \left(\frac{x}{x + 4}\right)$. This demonstrates the flexibility of the quotient property in handling both numerical and algebraic expressions.

It is important to avoid a common misconception: the quotient property applies only when division occurs inside the logarithm's argument. Expressions like $\log_b (m - n)$ cannot be simplified into $\log_b m - \log_b n$ because subtraction inside the argument does not correspond to any logarithmic property. Therefore, $\log_b (m - n)$ remains as is and cannot be broken down further using logarithmic rules.

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$_2x-$\log$_26$

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$_5x$

$\frac{\log_5125}{\log_5x}$

$$\log$_5x-3$

$\frac{\log_5x}{\log_5125}$

Problem

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

$$\log$_53$

$$\log$_5$\left$($\frac$13$\right$)$

$$\log$_516$

$-$\log$_516$

Problem

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

$$\log$_{10}4x$

$$\log$_{10}4x^2$

$$\log$_{10}32x$

$$\log$_{10}$\left$(8x^2-2x$\right$)$

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 pulling the exponent $n$ out in front of the logarithm as a multiplier, resulting in $n \times \log_b (m)$. This property is especially useful for simplifying complex logarithmic expressions and solving logarithmic equations.

For example, consider $\log_3 (7^2)$. Using the power property, this can be rewritten as $2 \times \log_3 (7)$. This transformation makes it easier to work with the logarithm, especially when combined with other logarithmic properties.

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 greater manipulation and simplification of logarithmic expressions.

Applying exponent rules alongside the power property can further enhance your ability to simplify logarithms. For instance, the square root of a number can be expressed as an exponent of one-half. Thus, $\log_2 (\sqrt{5})$ can be rewritten as $\log_2 (5^{\frac{1}{2}})$, 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 combining all logarithmic properties effectively in more advanced problems.

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 conversion uses the property that the nth root of a number is equivalent to raising that number to the power of 1/n. Therefore, the expression log₁₀ of the cube root of y⁴ becomes log₁₀ (y^4/3).

Next, applying the power property of logarithms, which states that log_b (a^c) = c × log_b (a), allows the exponent to be brought in front as a multiplier. Using this property, the expression simplifies to:

\[\frac{4}{3} \times \log_{10} y\]

This approach streamlines logarithmic expressions by converting roots to fractional exponents and then applying logarithmic properties to simplify the expression. Understanding how to manipulate exponents and logarithms in this way is essential for solving more complex logarithmic equations efficiently.

Problem

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

$5 the log base 9 of 4$

$5 the log base 4 of 9$

$4$\log$_95$

$4$\log$_59$

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$12$\log$_6m$

Problem

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

$x plus 1$

$2$\log$_2$\left$(x+1$\right$)$

$$\left$(x+1$\right$)$\log$_22$

$2+$\log$_2$\left$(x+1$\right$)$

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 a single logarithm as the sum or difference of multiple logarithms. For example, consider the expression $\log_{10}(3xy^2)$. Using the product property, this can be expanded into the sum of logarithms: $\log_{10}(3) + \log_{10}(x) + \log_{10}(y^2)$. To further expand, the power property allows the exponent to be brought in front as a multiplier, transforming $\log_{10}(y^2)$ into $2 \log_{10}(y)$. Thus, the fully expanded form is $\log_{10}(3) + \log_{10}(x) + 2 \log_{10}(y)$.

Conversely, condensing logarithmic expressions involves combining multiple logs into a single logarithm. This process typically begins with the power property, which converts coefficients in front of logs into exponents within the argument. For instance, the expression $4 \log_5(x) - \log_5(x + 2)$ can be condensed by first applying the power rule to get $\log_5(x^4) - \log_5(x + 2)$. Then, the quotient property combines the subtraction of logs into the logarithm of a quotient, resulting in $\log_5\left(\frac{x^4}{x + 2}\right)$. This is the most condensed form of the expression.

Understanding and applying these logarithmic properties—product, quotient, and power—enables the manipulation of complex logarithmic expressions for simplification or expansion. Mastery of these rules is essential for solving logarithmic equations, simplifying expressions, and analyzing logarithmic functions effectively.

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$_23+2$\log$_2a+4$\log$_2b-$\frac$12$\log$_25-2$\log$_2c$

$$\log$_23+2$\log$_2a+4$\log$_2b-$\frac$12$\log$_25+4$\log$_2c$

$$\log$_23+2$\log$_2a+4$\log$_2b-2$\log$_25-$\frac$12$\log$_2c$

$$\log$_23+2$\log$_2a+4$\log$_2b-2$\log$_25+$\frac$12$\log$_2c$

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{5x^2}{2}$

$\log\)_3\(\frac{5+2x}{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$_4$\left$($\frac{4a^3c}{\sqrt{b+1}$}$\right$)$

$$\log$_4$\left$($\frac{4a^3c}{\sqrt{b}$+1}$\right$)$

$$\log$_4$\left$($\frac{12a^{}$c}{$\sqrt{b}$+1}$\right$)$

Example

Expanding & Condensing Log Expressions Using Multiple Properties Example 3

Video duration:

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

When working with logarithmic expressions involving unknown bases, it is essential to rewrite the given logarithms in terms of known values using logarithmic properties. For example, if you are given values such as $\log_x 3 = -1.585$ and $\log_x 5 = -2.322$, you can approximate more complex logarithmic expressions by expressing them in terms of these known logs.

Consider the expression $\log_x \frac{45}{\sqrt{3}}$. 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 the known logs. Since $45 = 5 \times 3^2$, rewrite:

\[\log_x 45 = \log_x (5 \times 3^2)\]

Using the product property of logarithms, this becomes:

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

Apply the power property to bring down the exponent:

\[\log_x 5 + 2 \log_x 3\]

For the denominator, rewrite the square root as a fractional exponent:

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

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 value is approximately -4.700.

For a more complex example, consider $\log_x \frac{75}{3 \sqrt{5}}$. Begin by applying the quotient property:

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

Factor 75 as $3 \times 5^2$:

\[\log_x (3 \times 5^2)\]

Apply the product property:

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

For the denominator, use the product property again:

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

Distribute the negative sign:

\[- \log_x 3 - \frac{1}{2} \log_x 5\]

Combine all terms:

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

Substitute the known value for $\log_x 5$:

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

Rounded to three decimal places, the value is approximately -3.483.

These examples demonstrate how logarithmic properties—quotient, product, and power rules—allow complex logarithmic expressions to be rewritten in terms of simpler, known logarithms. This approach simplifies calculations and helps approximate values efficiently, especially when the base of the logarithm is unknown but certain logarithmic values are provided.

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 $\log b (m \cdot n) = \log b (m) + \log b (n)$ . This means that the logarithm of a product can be rewritten as the sum of the logarithms of the factors, provided the logs have the same base. 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 a single log with the product of their arguments. This property is useful for simplifying logarithmic expressions and solving equations involving logs.

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. You can apply this property to expand a single log with division inside into a subtraction of logs or condense a subtraction of logs into a single log with division inside. It is important to note that subtraction inside the argument (like $m - n$ ) cannot be simplified using this property.

The power property of logarithms states that $\log b (m^{n}) = n \log b (m)$ . This means that an exponent inside the argument of a logarithm can be pulled out front as a multiplier. Conversely, a coefficient multiplying a log can be moved inside as an exponent on the argument. This property is especially useful for rewriting roots (using fractional exponents) and simplifying expressions with powers. For example, $\log 2 (\sqrt{5})$ can be rewritten as $\frac{1}{2} \log 2 (5)$ .

No, logarithmic properties such as the product, quotient, and power properties do not apply to addition or subtraction inside the argument of a logarithm. For example, $\log b (m + n)$ cannot be rewritten as $\log b (m) + \log b (n)$ . Similarly, subtraction inside the argument cannot be separated into a difference of logs. These expressions cannot be simplified further using log properties and must be handled differently in algebraic manipulations.

To expand or condense complex logarithmic expressions, you apply the product, quotient, and power properties in sequence. When expanding, start by using the product and quotient properties to rewrite multiplication and division inside the argument as sums and differences of logs. Then apply the power property to move exponents out front as coefficients. For condensing, reverse the process: first apply the power property to move coefficients inside as exponents, then use the product and quotient properties to combine sums and differences of logs into single logarithms with multiplication or division inside. Always ensure the logs have the same base when combining. This systematic approach helps simplify or rewrite logarithmic expressions effectively.

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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 used to simplify logarithmic expressions?

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

Can logarithmic properties be applied to addition or subtraction inside the argument of a logarithm?

Can logarithmic properties be applied to addition or subtraction inside the argument of a logarithm?

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?

Your 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.
$\log_{3} (\frac{324}{4 \sqrt{3}}) = 4 \log_{3} 3 + \log_{3} 3 - \frac{1}{2} \log_{3} 4$