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Manipulating Logarithmic Functions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Properties of logarithms

What is the product property for logarithms?

The product property for logarithms states that
- $\log_{b} (x y) = \log_{b} x + \log_{b} y$
  - The logarithm of a product equals the sum of the logarithms
- This works in reverse too: a sum of logarithms (with the same base) can be combined into a single logarithm of a product
  - E.g. $\log_{3} 4 + \log_{3} 7 = \log_{3} (4 \cdot 7) = \log_{3} 28$
There is also a corresponding quotient property
- $\log_{b} (\frac{x}{y}) = \log_{b} x - \log_{b} y$
  - The logarithm of a quotient equals the difference of the logarithms
- This also works in reverse, to combine a difference of logarithms
  - E.g. $\ln 15 - \ln 3 = \ln \frac{15}{3} = \ln 5$
The product property has an important graphical implication
- Every horizontal dilation of a logarithmic function, $f (x) = \log_{b} (k x)$
- is equivalent to a vertical translation
- because $\log_{b} (k x) = \log_{b} k + \log_{b} x$
  - The constant $\log_{b} k$ acts as a vertical shift of $\log_{b} x$

What is the power property for logarithms?

The power property for logarithms states that
- $\log_{b} (x^{n}) = n \log_{b} x$
  - An exponent inside a logarithm can be brought out as a multiplier in front
  - E.g. $\log_{5} (x^{2}) = 2 \log_{5} x$
- This works in reverse too: a coefficient in front of a logarithm can be moved inside as an exponent
  - E.g. $3 \log_{2} x = \log_{2} (x^{3})$
  - E.g. $\frac{1}{2} \ln x = \ln (x^{1 / 2}) = \ln \sqrt{x}$

This also has an important graphical implication
- Raising the input of a logarithmic function to a power, $f (x) = \log_{b} (x^{k})$
- is equivalent to a vertical dilation by a factor of $k$
- because $\log_{b} (x^{k}) = k \log_{b} x$

What is the change of base property?

The change of base property for logarithms states that
- $\log_{b} x = \frac{\log_{a} x}{\log_{a} b}$
  - where $a > 0$ and $a \neq 1$
This allows you to convert a logarithm from one base to another
- E.g. $\log_{5} 12 = \frac{\ln 12}{\ln 5} = \frac{\log 12}{\log 5}$
- You can use any base for the conversion
  - $e$ and 10 are both commonly-used 'standard' bases
An important graphical implication of this is that all logarithmic functions are vertical dilations of each other
- Since $\log_{b} x = \frac{1}{\log_{a} b} \cdot \log_{a} x$
  - the function $\log_{b} x$ is just a constant multiple of $\log_{a} x$
- This means changing the base of a logarithmic function only stretches or compresses the graph vertically
  - It doesn't change the overall shape

How do you use these properties to rewrite expressions?

On the exam, you are frequently asked to combine multiple logarithmic terms into a single logarithm
- or to expand a single logarithm into multiple terms
For combining (multiple terms → single logarithm):
- First use the power property to move any coefficients inside as exponents
  - E.g. $2 \log_{3} x + \log_{3} 5 - 3 \log_{3} y = \log_{3} (x^{2}) + \log_{3} 5 - \log_{3} (y^{3})$
- Then use the product property (for addition) and quotient property (for subtraction) to combine into one logarithm
  - $= \log_{3} (5 x^{2}) - \log_{3} (y^{3})$
  - $= \log_{3} (\frac{5 x^{2}}{y^{3}})$

When combining terms, make sure all logarithms have the same base before applying the product or quotient properties
- If the bases differ, use the change of base property first
For expanding (single logarithm → multiple terms):
- Apply the product/quotient/power properties in reverse
- E.g. $\log_{2} (\frac{x^{3} \sqrt{y}}{z}) = \log_{2} (x^{3}) + \log_{2} (\sqrt{y}) - \log_{2} z = 3 \log_{2} x + \frac{1}{2} \log_{2} y - \log_{2} z$

Examiner Tips and Tricks

A common error with these types of questions is applying the power property incorrectly

Remember, $\log_{b} (x^{n}) = n \log_{b} x$
but $(\log_{b} x)^{n} \neq n \log_{b} x$

The exponent must be on the input, not on the whole logarithm.

Show every step clearly, both to assure you score all possible points, but also to help you avoid errors.

Worked Example

The function $j$ is given by $j (x) = 5 \log_{5} x + \log_{5} (3 x) - 2 \log_{5} (x^{2})$ .

Rewrite $j (x)$ as a single logarithm of the form $\log_{5} (expression)$ .

Answer:

First apply the power property to move the coefficients inside

$\begin{array}{rcl} j (x) & = & 5 \log_{5} x + \log_{5} (3 x) - 2 \log_{5} (x^{2}) \\ = & \log_{5} (x^{5}) + \log_{5} (3 x) - \log_{5} ((x^{2})^{2}) \\ = & \log_{5} (x^{5}) + \log_{5} (3 x) - \log_{5} (x^{4}) \end{array}$

Then use the product property to combine the first two terms

$= \log_{5} (x^{5} \cdot 3 x) - \log_{5} (x^{4}) = \log_{5} (3 x^{6}) - \log_{5} (x^{4})$

and the quotient property to combine into a single logarithm

$= \log_{5} (\frac{3 x^{6}}{x^{4}})$

Simplify by cancelling common factors

$j (x) = \log_{5} (3 x^{2})$

Worked Example

Let $a$ , $b$ , and $c$ be positive constants. Which of the following is equivalent to $\log_{10} (\frac{a^{2} c}{b^{3}})$ ?

(A) $\log_{10} (a^{2} + c) - \log_{10} (3 b)$

(B) $\frac{1}{2} \log_{10} a + \log_{10} c - \frac{1}{3} \log_{10} b$

(D) $2 \log_{10} a + \log_{10} c - 3 \log_{10} b$

Answer:

You could try combining the terms in all the answer options, to see which one is equal to $\log_{10} (\frac{a^{2} c}{b^{3}})$

But it is quicker to expand $\log_{10} (\frac{a^{2} c}{b^{3}})$ into separate terms

First use the quotient property

$\log_{10} (\frac{a^{2} c}{b^{3}}) = \log_{10} (a^{2} c) - \log_{10} (b^{3})$

Then use the product property

$= \log_{10} (a^{2}) + \log_{10} c - \log_{10} (b^{3})$

Then use the power property to bring the powers out front as multipliers

$= 2 \log_{10} a + \log_{10} c - 3 \log_{10} b$

That is option (D)

(D) $2 \log_{10} a + \log_{10} c - 3 \log_{10} b$

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Manipulating Logarithmic Functions (College Board AP® Precalculus): Study Guide

Properties of logarithms

What is the product property for logarithms?

What is the power property for logarithms?

What is the change of base property?

How do you use these properties to rewrite expressions?

Examiner Tips and Tricks

Worked Example

Worked Example

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Author: Roger B

Reviewer: Mark Curtis