College Algebra

Logarithmic Functions

Lesson

A logarithm is the inverse of an exponential. The expression $\log_b(x)$ asks the question:

\log_b(x) = y \iff b^y = x

In words: “ $\log_b(x)$ is the exponent you put on $b$ to get $x$ .”

Three facts that fall out of the definition:

$\log_b(1) = 0$ — because $b^0 = 1$ .
$\log_b(b) = 1$ — because $b^1 = b$ .
$\log_b(b^n) = n$ — the answer is the exponent.

The graph of $y = \log_b(x)$ has domain $(0,\,\infty)$ (you can’t take a logarithm of zero or a negative number), range all real numbers, and a vertical asymptote at $x = 0$ . Two shorthand notations to know: $\log(x)$ with no base means base 10 (the common log), and $\ln(x)$ means base $e$ (the natural log).

To evaluate by hand: rewrite the input as a power of the base.

Worked example 1

\log_2(32) = ?

Rewrite 32 as a power of 2: $32 = 2^5$ . So:

\log_2(32) = \log_2(2^5) = 5

Worked example 2

\log_8(4) = ?

Both 8 and 4 are powers of 2, so write them with that common base: $8 = 2^3$ and $4 = 2^2$ . The question is: what exponent $y$ makes $(2^3)^y = 2^2$ ? That gives $3y = 2$ , so $y = \tfrac{2}{3}$ .

\log_8(4) = \tfrac{2}{3}

How to type your answer

Type a single number — the exponent. Use a slash for fractions. Examples: 5, -2, 1/2, 2/3.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\log_2(8) = ?

Problem 2

\log_3(9) = ?

Problem 3

\log_5(25) = ?

Problem 4

\log_{10}(1000) = ?

Practice

Standard problems matching the lesson.

Problem 5

\log_2(16) = ?

Problem 6

\log_2(1) = ?

Problem 7

\log_3(27) = ?

Problem 8

\log_4(64) = ?

Problem 9

\log_2\!\left(\tfrac{1}{4}\right) = ?

Problem 10

\log_5\!\left(\tfrac{1}{25}\right) = ?

Problem 11

\log_{10}(0.01) = ?

Problem 12

\log_2(32) = ?

Problem 13

\log_3(81) = ?

Problem 14

\log_2\!\left(\tfrac{1}{8}\right) = ?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\log_4(2) = ?

Problem 16

\log_8(2) = ?

Problem 17

\log_9(3) = ?

Problem 18

\log_{27}(3) = ?

Problem 19

\log_4(8) = ?

Problem 20

\log_8(4) = ?

Problem 21

\log_{25}(125) = ?

Problem 22

\log_2(\sqrt{2}) = ?