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Logarithmic Expressions (College Board AP® Precalculus): Study Guide

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Definition of a logarithm

What is a logarithm?

A logarithm answers the question: "What power must I raise a base to, in order to get a particular value?"
In the expression $\log_{b} c$
- $b$ is the base
  - and $c$ is the 'particular value'
- So the value of that expression represents
  - the exponent (power) that the base $b$ must be raised to
  - in order to produce the value $c$
- The base $b$ must satisfy $b > 0$ and $b \neq 1$
- The value $c$ must be positive
  - you cannot take the logarithm of zero or a negative number
In other words
- $\log_{b} c = a$
  - means exactly the same thing as $b^{a} = c$
- These two forms (logarithmic form and exponential form) are just two equivalent ways of writing the same relationship
Here are some examples:

Logarithmic form	Exponential form	In words
$\log_{2} 8 = 3$	$2^{3} = 8$	"2 raised to the power 3 gives 8"
$\log_{5} 25 = 2$	$5^{2} = 25$	"5 raised to the power 2 gives 25"
$\log_{10} 1000 = 3$	$10^{3} = 1000$	"10 raised to the power 3 gives 1000"
$\log_{3} \frac{1}{9} = - 2$	$3^{- 2} = \frac{1}{9}$	"3 raised to the power $- 2$ gives $\frac{1}{9}$ "

Examiner Tips and Tricks

Being able to move confidently between logarithmic form ( $\log_{b} c = a$ ) and exponential form ( $b^{a} = c$ ) is essential.

Many problems become much easier when you convert to whichever form is more convenient

What is the common logarithm?

When no base is written, the logarithm is understood to have base 10
- This is called the common logarithm
E.g. $\log 100$ means the same thing as $\log_{10} 100$
- and it is equal to 2 because $10^{2} = 100$
The common logarithm appears frequently on calculators and in applications

How do you evaluate logarithmic expressions?

Some logarithmic values can be worked out using basic arithmetic
- i.e. by thinking about what power of the base gives the desired value
- E.g. $\log_{2} 32$
  - since $2^{5} = 32$ , it follows that $\log_{2} 32 = 5$
- E.g. $\log_{4} \frac{1}{16}$
  - since $4^{- 2} = \frac{1}{4^{2}} = \frac{1}{16}$ , it follows that $\log_{4} \frac{1}{16} = - 2$
- E.g. $\log_{9} 3$
  - since $9^{\frac{1}{2}} = \sqrt{9} = 3$ , it follows that $\log_{9} 3 = \frac{1}{2}$
Other values cannot be evaluated exactly by hand
- but can be estimated using technology (a calculator)
- E.g. $\log_{2} 5$
  - There is no integer or simple fraction $a$ such that $2^{a} = 5$ exactly
  - But a calculator will give you the value
    - $\log_{2} 5 = 2.321928 . . .$

Examiner Tips and Tricks

Don't forget that logarithms can give negative answers (when the value is a reciprocal power of the base) and fractional answers (when the value is a root of the base).

Worked Example

Evaluate each of the following logarithmic expressions, without a calculator where possible.

(a) $\log_{3} 81$

Answer:

You need the power of 3 that gives 81

$3^{1} = 3$ , $3^{2} = 9$ , $3^{3} = 27$ , $3^{4} = 81$

$\log_{3} 81 = 4$

(b) $\log_{10} 0.01$

Answer:

You need the power of 10 that gives 0.01

$0.01 = \frac{1}{100} = \frac{1}{10^{2}} = 10^{- 2}$

$\log_{10} 0.01 = - 2$

Answer:

You need the power of 8 that gives 2

$8^{a} = 2$

Substitute $8 = 2^{3}$ and use laws of exponents

${(2^{3})}^{a} = 2^{1}$

$2^{3 a} = 2^{1}$

$3 a = 1$

$a = \frac{1}{3}$

$\log_{8} 2 = \frac{1}{3}$

(d) $\log_{5} 12$

Answer:

You need the power of 5 that gives 12

$5^{1} = 5$ and $5^{2} = 25$ , so the answer is somewhere between 1 and 2
but there is no exact value you can find by hand.

Using your calculator

$\log_{5} 12 = 1.543959 . . .$

Round to 3 decimal places

$\log_{5} 12 = 1.544 (3 d . p .)$

Logarithmic scales

What is a logarithmic scale?

On a standard (linear) scale, each unit represents the same additive change
- E.g. the marks might be at $0, 1, 2, 3, 4, \dots$
  - with equal spacing between them
On a logarithmic scale, each unit represents the same multiplicative change
- specifically, a multiplication by the base of the logarithm
E.g. on a base-10 logarithmic scale
- the marks might be at
  - $10^{0} = 1$ , $10^{1} = 10$ , $10^{2} = 100$ , $10^{3} = 1000$ , $\dots$
- Each step along the scale multiplies the value by 10, rather than adding a fixed amount
Logarithmic scales are useful for displaying data that would otherwise span a very wide range of values
Examples of commonly-used logarithmic scales include:
- earthquake magnitudes (the Richter scale)
- sound intensity (decibels)
- and the pH scale in chemistry

How does a logarithmic scale work?

On a logarithmic scale with base $b$
- the position of a value on the scale is its logarithm base $b$
E.g. on a base-10 scale
- the value 1 sits at position 0 (since $\log_{10} 1 = 0$ )
- the value 10 sits at position 1
- the value 100 sits at position 2
- etc.
Equal spacing on the scale corresponds to equal ratios in the actual values
- not equal differences
E.g. the distance from 10 to 100 on a base-10 log scale is the same as the distance from 100 to 1000
- both represent a multiplication by 10
Values that are close together on a linear scale (like 0.01 and 0.001) may be far apart on a logarithmic scale
- and values that seem vastly different on a linear scale (like 1000 and 10,000) may be close together on a logarithmic scale

Comparison of linear scale with equal differences ranging -3 to 6, and logarithmic scale with equal ratios from 10^-3 to 10^6. — **Linear scale (left) versus logarithmic scale (right)**

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Logarithmic Expressions (College Board AP® Precalculus): Study Guide

Definition of a logarithm

What is a logarithm?

Examiner Tips and Tricks

What is the common logarithm?

How do you evaluate logarithmic expressions?

Examiner Tips and Tricks

Worked Example

Logarithmic scales

What is a logarithmic scale?

How does a logarithmic scale work?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis