Write the log expression as a single log. log ⁡ 2 1 9 x + 2 log ⁡ 2 3 x \log_2\frac{1}{9x}+2\log_23x lo g 2 9 x 1 + 2 lo g 2 3 x

log ⁡ 2 1 3 x \log_2\frac{1}{3x} lo g 2 3 x 1

log ⁡ 2 3 x \log_23x lo g 2 3 x

Write the log expression as a single log. ln ⁡ 3 x y + 2 ln ⁡ 2 y − ln ⁡ 4 x \ln\frac{3x}{y}+2\ln2y-\ln4x ln y 3 x + 2 ln 2 y − ln 4 x

ln ⁡ ( 3 y ) \ln\left(3y\right) ln ( 3 y )

ln ⁡ 3 x y 4 \ln\frac{3xy}{4} ln 4 3 x y

ln ⁡ ( 12 x 2 ) \ln\left(12x^2\right) ln ( 12 x 2 )

ln ⁡ ( 3 2 ) \ln\left(\frac32\right) ln ( 2 3 )

Write the single logarithm as a sum or difference of logs. log ⁡ 3 ( x 9 y 2 ) \log_3\left(\frac{\sqrt{x}}{9y^2}\right) lo g 3 ( 9 y 2 x <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> )

1 2 log ⁡ 3 x − 2 − 2 log ⁡ 3 y \frac12\log_3x-2-2\log_3y

2 l o g 3 x − 2 − l o g 3 9 y 2log_3x-2-log_39y

1 2 l o g 3 x + 2 l o g 3 3 y \frac12log_3x+2log_33y

1 2 l o g 3 x − 2 l o g 3 9 y \frac12log_3x-2log_39y

Write the single logarithm as a sum or difference of logs. log ⁡ 5 ( 5 ( 2 x + 3 ) 2 x 3 ) \log_5\left(\frac{5\left(2x+3\right)^2}{x^3}\right) lo g 5 ( x 3 5 ( 2 x + 3 ) 2 )

1 + 2 log ⁡ 5 ( 2 x + 3 ) − 3 log ⁡ 5 x 1+2\log_5\left(2x+3\right)-3\log_5x 1 + 2 lo g 5 ( 2 x + 3 ) − 3 lo g 5 x

5 + 2 log ⁡ 5 ( 2 x + 3 ) − log ⁡ 5 3 x 5+2\log_5\left(2x+3\right)-\log_53x 5 + 2 lo g 5 ( 2 x + 3 ) − lo g 5 3 x

2 log ⁡ 5 ( 2 x + 3 ) − 3 log ⁡ 5 x 2\log_5\left(2x+3\right)-3\log_5x 2 lo g 5 ( 2 x + 3 ) − 3 lo g 5 x

log ⁡ 5 ( 2 x + 3 ) − log ⁡ 5 x \log_5\left(2x+3\right)-\log_5x lo g 5 ( 2 x + 3 ) − lo g 5 x

0. Functions

Properties of Logarithms

0. Functions

Properties of Logarithms: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Understanding logarithms involves recognizing their properties, such as the inverse relationship with exponentials. Key rules include the product rule, which states that $\log (b (m n)) = \log (b (m) + log(b(n))$ , and the quotient rule, which states that $\log (b (m/n))=\log(b(m)-log(b(n))$ . Additionally, changing the base of a logarithm allows for easier calculations, using the formula $\log (b (m)=log(a(m)/log(a(b))$ .

concept

Evaluate Logarithms

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Evaluate Logarithms Video Summary

Understanding logarithms is essential as they serve as the inverse of exponential functions. This relationship allows us to evaluate logarithmic expressions without a calculator. For instance, consider the expression log₂(√[3]{2}). By recognizing that the cube root of 2 can be expressed as an exponent, we rewrite it as log₂(2^1/3). Utilizing the inverse property of logarithms, where log_b(b^x) = x, we find that this simplifies to 1/3.

Several key properties of logarithms facilitate these evaluations. The first is the inverse property, which states that if the base of the logarithm matches the base of the exponent, they cancel each other out. For example, log₂(2³) = 3 and 2^log₂(2) = 1. This property holds true for any base, such as log_e(e^x) = x.

Another important property is that the logarithm of a number to its own base equals 1. For instance, log₂(2) = 1 because 2¹ = 2. Additionally, the logarithm of 1 in any base is always 0, as b⁰ = 1 for any base b.

To illustrate these properties, consider the natural logarithm of 1, which is log_e(1) = 0. Similarly, the common logarithm log(10) simplifies to log₁₀(10) = 1. For a more complex example, log₅(1/5) can be rewritten as log₅(5^-1), which simplifies to -1.

By mastering these properties and techniques, evaluating logarithmic expressions becomes a straightforward process, allowing for deeper understanding and application in various mathematical contexts.

Problem

Evaluate the given logarithm.
$\log_77^{0.3}$

$1.79$

$7$

$1$

$0.3$

Problem

Evaluate the given logarithm.
$\frac32\log1$

$\frac32$

$0$

$1$

$10$

Problem

Evaluate the given logarithm.
$\log_9\frac{1}{81}$

$81$

$2$

$-2$

$9$

concept

Product, Quotient, and Power Rules of Logs

Video duration:

Product, Quotient, and Power Rules of Logs Video Summary

When expanding logarithmic expressions, it's essential to understand the properties of logarithms that correspond to the rules of exponents. This connection allows us to manipulate logarithmic expressions effectively without needing to memorize entirely new rules. Here’s a concise overview of these properties:

The product rule states that when you have a logarithm of a product, such as \log_b(m \cdot n), you can expand it into the sum of two logarithms: \log_b(m) + \log_b(n). This mirrors the exponent rule where multiplying two numbers with the same base results in adding their exponents.

Similarly, the quotient rule applies when dealing with division in logarithms. For a logarithm of a quotient, like \log_b\left(\frac{m}{n}\right), it can be expressed as the difference of two logarithms: \log_b(m) - \log_b(n). This reflects the exponent rule where dividing two numbers with the same base leads to subtracting their exponents.

Lastly, the power rule indicates that when you have a logarithm of a number raised to a power, such as \log_b(m^n), you can bring the exponent in front of the logarithm: n \cdot \log_b(m). This is akin to the exponent rule where raising a power to another power results in multiplying the exponents.

For example, if you encounter \log_2(3x), you can apply the product rule to expand it to \log_2(3) + \log_2(x). In another case, \log_5\left(\frac{5}{y}\right) can be simplified using the quotient rule to \log_5(5) - \log_5(y). Lastly, for \ln(7^2), you would use the power rule to rewrite it as 2 \cdot \ln(7).

Understanding these properties not only aids in expanding logarithmic expressions but also enhances your overall grasp of logarithmic and exponential relationships, making problem-solving more intuitive.

concept

Expand & Condense Log Expressions

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Expand & Condense Log Expressions Video Summary

Understanding logarithmic expressions involves mastering the rules for both expanding and condensing logs. When expanding a logarithmic expression, such as log₂(3xy²), the product rule is essential. This rule states that the logarithm of a product can be expressed as the sum of the logarithms of the individual factors. Therefore, we can expand the expression as follows:

log₂(3xy²) = log₂(3) + log₂(x) + log₂(y²).

Next, we can apply the power rule, which allows us to move the exponent in front of the logarithm. For the term log₂(y²), this becomes 2 * log₂(y). Thus, the fully expanded expression is:

log₂(3) + log₂(x) + 2 * log₂(y).

On the other hand, condensing logarithmic expressions requires careful attention to the base and the operations involved. For instance, when condensing 2 * ln(x) - ln(x + 2), we first apply the power rule to the first term, transforming it into ln(x²). The expression now reads:

ln(x²) - ln(x + 2).

Since we are subtracting logarithms, we can use the quotient rule, which states that the logarithm of a quotient can be expressed as the difference of the logarithms. Therefore, we condense the expression to:

ln( (x²) / (x + 2) ).

By mastering these rules—product, power, and quotient—you can effectively expand and condense any logarithmic expression you encounter. Practice with various examples will enhance your understanding and proficiency in manipulating logarithmic forms.

Problem

Write the log expression as a single log.
$\log_2\frac{1}{9x}+2\log_23x$

$\log_2x$

$\log_2\frac{1}{3x}$

$\log_21$

$\log_23x$

Problem

Write the log expression as a single log.
$\ln\frac{3x}{y}+2\ln2y-\ln4x$

$\ln\frac{3xy}{4}$

$\ln\left(12x^2\right)$

$\ln\left(\frac32\right)$

$\ln\left(3y\right)$

Problem

Write the single logarithm as a sum or difference of logs.
$\log_3\left(\frac{\sqrt{x}}{9y^2}\right)$

$2 l o g_{3} x - 2 - l o g_{3} 9 y$

$\frac{1}{2} \log_{3} x - 2 - 2 \log_{3} y$

$one half l o g sub 3 of x plus 2 l o g sub 3 of 3 y$

$one half l o g sub 3 of x minus 2 l o g sub 3 of 9 y$

Problem

Write the single logarithm as a sum or difference of logs.
$\log_5\left(\frac{5\left(2x+3\right)^2}{x^3}\right)$

$5+2\log_5\left(2x+3\right)-\log_53x$

$2\log_5\left(2x+3\right)-3\log_5x$

$1+2\log_5\left(2x+3\right)-3\log_5x$

$\log_5\left(2x+3\right)-\log_5x$

concept

Change of Base Property

Video duration:

Change of Base Property Video Summary

When evaluating logarithms, especially those with unconventional bases like π, it can be beneficial to change the base to a more manageable one, such as 10 or e. This process allows for easier calculations using a calculator. The change of base formula is essential for this purpose: if you have a logarithm of the form log_b(m), you can convert it to a new base a using the formula:

log_a(m) = log_a(m) / log_a(b)

In this formula, the original logarithm's argument m is placed in the numerator, while the original base b is in the denominator. For example, to evaluate log₅(2) using base 10, you would express it as:

log₅(2) = log(2) / log(5)

Calculators typically have buttons for common logarithms (base 10) and natural logarithms (base e), making these bases the most convenient for quick evaluations. For instance, if you need to evaluate log₇(31), you can convert it to:

log₇(31) = log(31) / log(7)

After entering this into a calculator, you would find that log₇(31) ≈ 1.76.

Similarly, for log_π(9), you can change the base to 10:

log_π(9) = log(9) / log(π)

Calculating this yields approximately 1.92. If you choose to use natural logarithms instead, the expression becomes:

log_π(9) = ln(9) / ln(π)

This will also result in approximately 1.92, demonstrating that the base change does not affect the final result.

In another example, evaluating log_√3(e) using natural logarithms gives:

log_√3(e) = ln(e) / ln(√3)

Since ln(e) = 1, this simplifies to:

log_√3(e) = 1 / ln(√3)

Calculating this will yield approximately 1.82.

Understanding how to change the base of logarithms is a powerful tool that allows for flexibility in calculations, ensuring that you can evaluate logarithmic expressions efficiently regardless of their original base.

Problem

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.
$\log_317$

$0.39$

$2.58$

$1.23$

$0.48$

Problem

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.
$\log_967$

$1.91$

$0.52$

$0.95$

$1.83$

Problem

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.
$\log_841$

$1.61$

$0.9$

$0.56$

$1.79$

Problem

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.
$\log_23789$

$0.08$

$11.89$

$3.58$

$0.30$

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Properties of Logarithms quiz #1
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Here’s what students ask on this topic:

What are the properties of logarithms?

The properties of logarithms include:

Product Rule: $\log_{b} (m n) = \log_{b} (m) + \log_{b} (n)$
Quotient Rule: $\log_{b} (\frac{m}{n}) = \log_{b} (m) - \log_{b} (n)$
Power Rule: $\log_{b} (m^{n}) = n \log_{b} (m)$
Change of Base Formula: $\log_{b} (m) = \frac{\log_{a}}{(}$

How do you expand logarithmic expressions?

To expand logarithmic expressions, use the properties of logarithms:

Product Rule: $\log_{b} (m n) = \log_{b} (m) + \log_{b} (n)$
Quotient Rule: $\log_{b} (\frac{m}{n}) = \log_{b} (m) - \log_{b} (n)$
Power Rule: $\log_{b} (m^{n}) = n \log_{b} (m)$

For example, to expand $\log_{2} (3xy y^{2})$ , use the product rule: $\log_{2} (3) + \log_{2} (x) + \log_{2} (y^{2})$ . Then apply the power rule: $\log_{2} (3) + \log_{2} (x) + 2 \log_{2} (y)$ .

How do you condense logarithmic expressions?

To condense logarithmic expressions, use the properties of logarithms in reverse:

Product Rule: $\log_{b} (m) + \log_{b} (n) = \log_{b} (m n)$
Quotient Rule: $\log_{b} (m) - \log_{b} (n) = \log_{b} (\frac{m}{n})$
Power Rule: $n \log_{b} (m) = \log_{b} (m^{n})$

For example, to condense $2 \ln (x) - \ln (x + 2)$ , first apply the power rule: $\ln (x^{2) - \ln (x + 2)}$ . Then apply the quotient rule: $\ln (\frac{x^{2}}{+})$ .

How do you change the base of a logarithm?

To change the base of a logarithm, use the change of base formula:

\log_{b} (m) = \frac{\log_{a}}{(}

For example, to change $\log_{7} (31)$ to base 10, use:

\frac{\log}{(}

Then, you can use a calculator to find the values of $\log (31)$ and $\log (7)$ and divide them to get the result.

How do you evaluate logarithms without a calculator?

To evaluate logarithms without a calculator, use the properties of logarithms:

Inverse Property: $\log_{b} (b^{x}) = x$
Log of 1: $\log_{b} (1) = 0$
Log of the Base: $\log_{b} (b) = 1$

For example, to evaluate $\log_{2} (\sqrt[3]{2})$ , rewrite it as $\log_{2} (2^{\frac{1}{3})}$ . Using the inverse property, this simplifies to $\frac{1}{3}$ .

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Additional resources for Properties of Logarithms

PRACTICE PROBLEMS AND ACTIVITIES (10)

Properties of Logarithms: Videos & Practice Problems

Evaluate Logarithms

Evaluate Logarithms Video Summary

Evaluate the given logarithm.log770.3\log_77^{0.3}log770.3

Evaluate the given logarithm.32log1\frac32\log123log1

Evaluate the given logarithm.log9181\log_9\frac{1}{81}log9811

Product, Quotient, and Power Rules of Logs

Product, Quotient, and Power Rules of Logs Video Summary

Expand & Condense Log Expressions

Expand & Condense Log Expressions Video Summary

Write the log expression as a single log.log219x+2log23x\log_2\frac{1}{9x}+2\log_23xlog29x1+2log23x

Write the log expression as a single log.ln3xy+2ln2y−ln4x\ln\frac{3x}{y}+2\ln2y-\ln4xlny3x+2ln2y−ln4x

Write the single logarithm as a sum or difference of logs.log3(x9y2)\log_3\left(\frac{\sqrt{x}}{9y^2}\right)log3(9y2x)

Write the single logarithm as a sum or difference of logs.log5(5(2x+3)2x3)\log_5\left(\frac{5\left(2x+3\right)^2}{x^3}\right)log5(x35(2x+3)2)

Change of Base Property

Change of Base Property Video Summary

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.log317\log_317log317

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.log967\log_967log967

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.log841\log_841log841

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log. log23789\log_23789log23789

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Go over this topic definitions with flashcards

Here’s what students ask on this topic:

What are the properties of logarithms?

How do you expand logarithmic expressions?

How do you condense logarithmic expressions?

How do you change the base of a logarithm?

How do you evaluate logarithms without a calculator?

Your Calculus tutors

Evaluate the given logarithm.
$\log_77^{0.3}$

Evaluate the given logarithm.
$\frac32\log1$

Evaluate the given logarithm.
$\log_9\frac{1}{81}$

Write the log expression as a single log.
$\log_2\frac{1}{9x}+2\log_23x$

Write the log expression as a single log.
$\ln\frac{3x}{y}+2\ln2y-\ln4x$

Write the single logarithm as a sum or difference of logs.
$\log_3\left(\frac{\sqrt{x}}{9y^2}\right)$

Write the single logarithm as a sum or difference of logs.
$\log_5\left(\frac{5\left(2x+3\right)^2}{x^3}\right)$

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.
$\log_317$

Evaluate the given logarithm using the change of base formula and a calculator. Use the common log.
$\log_967$

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.
$\log_841$

Evaluate the given logarithm using the change of base formula and a calculator. Use the natural log.
$\log_23789$