Logarithmic Expressions

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

A logarithm answers the question: what power of the base $b$ gives $x$ ? Symbolically, $\log_b x = y$ iff $b^y = x$ . The common log is base 10, the natural log is base $e$ . The argument $x$ must be positive — $\log_b x$ is undefined for $x \leq 0$ (and the base must satisfy $b > 0, b \neq 1$ ).

Definition

Equivalence

\log_b x = y \iff b^y = x \quad (b > 0, b \neq 1, x > 0)

Special values

\log_b 1

= 0

\log_b b

= 1

\log_b(b^k)

= k

Inverse identity

b^{\log_b x} = x

Caution: Logarithms of zero or negative numbers are undefined. Always check the argument is positive before evaluating.

AP problem types

Type 1

Direct evaluation

Rewrite the argument as an integer power of the base, then read off the exponent.

Example 1

Evaluate

\log_2 32

. (A)

4

(B)

5

(C)

6

(D)

32

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Type 2

Convert exponential to logarithmic form

Rewrite $b^y = x$ as $y = \log_b x$ , isolating the exponent on the right.

Example 2

Rewrite

3^x = 7

in logarithmic form. (A)

x = \log_7 3

(B)

x = \log_3 7

(C)

7 = \log_x 3

(D)

x = 7 \log 3

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Type 3

Domain and special-case identities

Apply $\log_b 1 = 0$ , $\log_b b = 1$ , $\log_b(b^k) = k$ . Reject negative arguments as undefined.

Example 3

Which of the following is undefined? (A)

\log_2 32

(B)

\log_3(1/27)

(C)

\log_2(-8)

(D)

\ln(e^4)

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