Logarithmic Expressions — AP® Precalculus Notes

Exponential functions answer the question "what is $2^5$ ?". Often we need the reverse: " $2$ to what power is $32$ ?". The answer is a logarithm. This page defines logarithms, shows how to find exact values, introduces the common and natural logarithms, and explains logarithmic scales.

A logarithm is an exponent

For a base $b > 0$ with $b \ne 1$ and a positive number $c$ :

$\log_b c = a \quad \text{means} \quad b^a = c.$

Read $\log_b c$ as "log base $b$ of $c$ ". It is the exponent that turns $b$ into $c$ . Because $f(x) = b^x$ is invertible, every positive $c$ comes from exactly one exponent, so this exponent is well defined.

Exponential form	Logarithmic form	In words
$2^5 = 32$	$\log_2 32 = 5$	$2$ needs the exponent $5$ to make $32$
$10^{-2} = 0.01$	$\log 0.01 = -2$	a number below $1$ has a negative logarithm
$7^0 = 1$	$\log_7 1 = 0$	every base needs the exponent $0$ to make $1$
$e^1 = e$	$\ln e = 1$	a base needs the exponent $1$ to make itself

Finding exact values

Write the input as a power of the base. Then the logarithm is that power.

Worked example 1. Find $\log_3 \frac{1}{9}$ and $\log_4 8$ .

$\frac{1}{9} = \frac{1}{3^2} = 3^{-2}$ , so $\log_3 \frac{1}{9} = -2$ .

For $\log_4 8$ , $8$ is not a whole-number power of $4$ . Write both as powers of $2$ : $4 = 2^2$ and $8 = 2^3$ . We need $4^a = 8$ , that is $2^{2a} = 2^3$ , so $2a = 3$ and $\log_4 8 = \frac{3}{2}$ .

Worked example 2. Solve $\log_2 x = 7$ .

Rewrite in exponential form: $x = 2^7 = 128$ .

Common and natural logarithms

Two bases are used so often that they have their own names and calculator keys.

The common logarithm has base $10$ and is written $\log c$ , with no base shown: $\log 1{,}000 = 3$ .
The natural logarithm has base $e \approx 2.718$ and is written $\ln c$ : $\ln e^4 = 4$ and $\ln 1 = 0$ .

Mixing them up is a common calculator slip: $\log 50 \approx 1.699$ but $\ln 50 \approx 3.912$ .

Logarithms that are not whole numbers

Most logarithms are not whole numbers, but you can always say which two whole numbers they lie between.

Worked example 3. Between which consecutive integers is $\log_3 50$ ?

List powers of $3$ : $3^3 = 27$ and $3^4 = 81$ . Since $27 < 50 < 81$ , the exponent is between $3$ and $4$ .

A calculator gives $\log_3 50 \approx 3.561$ , which agrees.

What a logarithm can and cannot take

With a positive base, every power $b^a$ is positive. So $\log_b c$ exists only when $c > 0$ : $\log_2 0$ and $\log_2(-4)$ are undefined. The value of a logarithm, though, can be any real number, including $0$ and negative numbers: $\log_2 \frac{1}{2} = -1$ .

Logarithmic scales

Some quantities span many powers of ten: sound intensity, the strength of ground shaking, the acidity of a solution, the number of users of an app. A logarithmic scale places each number at the distance given by its logarithm. Then $1$ , $10$ , $100$ and $1{,}000$ are equally spaced, and each equal step multiplies by the same factor.

A number line on a logarithmic scale: 1, 10, 100, 1,000 and 10,000 equally spaced, with the values 50 and 2,000 marked. — **Figure 1.** On a logarithmic scale, $50$ sits at $\log 50 \approx 1.7$ steps from $1$ and $2{,}000$ at $\log 2{,}000 \approx 3.3$ steps. LevlPrep original diagram.

Worked example 4. On a scale with reading $M = \log A$ , one reading is $6$ and another is $4$ . Compare the amounts $A$ .

$M = 6$ means $A = 10^6$ ; $M = 4$ means $A = 10^4$ .

The ratio is $\frac{10^6}{10^4} = 10^2 = 100$ . A difference of $2$ on the scale is a factor of $100$ , not of $2$ .

The rule to remember: on a logarithmic scale, differences in position stand for ratios of values.