Algebra II

Logarithm Properties and Evaluating

Lesson

A logarithm is the inverse of an exponent. $\log_b(x) = y$ means “b raised to what power gives x?” In other words:

\log_b(x) = y \;\Longleftrightarrow\; b^y = x

Read $\log_b(x)$ as “log base b of x.” Common bases are $\log_{10}$ (often written just $\log$ ) and $\ln$ (natural log, base $e$ ).

Evaluating logs by definition

$\log_2(8) = 3$ because $2^3 = 8$ .

Three log properties

Product rule

\log_b(MN) = \log_b(M) + \log_b(N)

Quotient rule

\log_b\!\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)

Power rule

\log_b(M^p) = p \cdot \log_b(M)

Two more useful facts: $\log_b(b) = 1$ and $\log_b(1) = 0$ .

Worked example 1

Evaluate $\log_3(81)$ .

Ask: 3 to what power gives 81? $3^4 = 81$ , so the answer is 4.

Worked example 2

Use properties to evaluate $\log_2(32) - \log_2(4)$ .

By the quotient rule:

\log_2\!\left(\frac{32}{4}\right) = \log_2(8) = 3

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\log_2(4)

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(8)

Problem 6

\log_2(16)

Problem 7

\log_3(27)

Problem 8

\log_3(81)

Problem 9

\log_4(64)

Problem 10

\log_2(1)

Problem 11

\log_7(7)

Problem 12

\log_2(8) + \log_2(4)

Problem 13

\log_3(27) - \log_3(3)

Problem 14

\log_2(64) - \log_2(8)

Problem 15

\log_5(125) + \log_5(5)

Problem 16

\log_2(2^5)

Challenge

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

Problem 17

\log_2(\tfrac{1}{4})

Problem 18

\log_3(\tfrac{1}{27})

Problem 19

\log_4(2)

Problem 20

\log_9(3)

Problem 21

\log_2(32) + \log_2(\tfrac{1}{8})

Problem 22

\log_5(125) - \log_5(25) + \log_5(5)

Practice

Standard problems matching the lesson.

Problem 23

Evaluate log(8) + log(125), all logs base 10.

Problem 24

Evaluate log(100³), log base 10.

Challenge

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

Problem 25

Find log(10000) − log(100), log base 10.

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