Logarithmic Expressions — AP® Precalculus Lesson

Part 1 · Hook

Why this matters

The quietest sound you can hear and a very loud rock concert differ in intensity by a factor of about

100

billion. No ordinary ruler shows both. Counting powers of ten instead, the gap becomes

11

steps. A logarithm is that count: the exponent a base needs to reach a number.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is $2^{-3}$ ?

$\frac{1}{8}$
$-8$
$-6$
$\frac{1}{6}$

Show the answer

A negative exponent means a reciprocal: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$ .

Correct: $\frac{1}{8}$ :
$-8$ :
$-6$ :
$\frac{1}{6}$ :

2. $f(x) = 2^x$ is invertible. Since $f(5) = 32$ , what is $f^{-1}(32)$ ?

$5$
$32$
$\frac{1}{32}$
$16$

Show the answer

The inverse sends the output $32$ back to the input $5$ .

Correct: $5$ :
$32$ :
$\frac{1}{32}$ :
$16$ :

3. Which number is $e$ closest to?

$2.718$
$3.142$
$1.618$
$2.000$

Show the answer

The natural base $e$ is about $2.718$ .

Correct: $2.718$ :
$3.142$ :
$1.618$ :
$2.000$ :

Part 4 · See it

See it first

Graph of y = 2 to the x through (1, 2), (2, 4) and (3, 8); the input that gives the output 8 is 3, which is log base 2 of 8. — On $y = 2^x$ , the output $8$ comes from the input $3$ . That input is the logarithm: $\log_2 8 = 3$ . LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Ask which power of $b$ gives $c$ the answer $a$ is $\log_b c$ , so $\log_b c = a$ means $b^a = c$
Use base $10$ or base $e$ you get the common logarithm $\log c$ or the natural logarithm $\ln c$
$c$ lies between two powers of $b$ $\log_b c$ lies between their exponents
Every power $b^a$ with $b > 0$ is positive $\log_b c$ exists only for $c > 0$ , though its value can be negative
Space numbers by their logarithmsequal distances mean equal factors: a logarithmic scale

Part 6 · Key ideas

Key ideas

$\log_b c = a$ means $b^a = c$ , for $b > 0$ , $b \ne 1$ , $c > 0$ . A logarithm is an exponent.
Common logarithm: $\log c = \log_{10} c$ . Natural logarithm: $\ln c = \log_e c$ .
Exact values come from writing $c$ as a power of $b$ : $\log_4 8 = \frac{3}{2}$ because $8 = 4^{3/2}$ .
On a logarithmic scale, each equal step multiplies the value by the same factor.

Part 7 · Misconception

A common mistake

The wrong idea: $\log_3 81 = 27$ , because $81 \div 3 = 27$ .

What actually happens: A logarithm gives the exponent, not a quotient: $3^4 = 81$ , so $\log_3 81 = 4$ .

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Powers of $3$

Values of 3 to the power x for x from -2 to 4
$x$	-2	-1	0	1	2	3	4
$3^x$	$\frac{1}{9}$	$\frac{1}{3}$	1	3	9	27	81

1. Use the table to find $\log_3 81$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$\log_3 81$ asks: $3$ to what power is $81$ ? The table shows $3^4 = 81$ , so $\log_3 81 = 4$ .

Answer: 4

2. Use the table to find $\log_3 \frac{1}{9}$ .

$-2$
$2$
$\frac{1}{9}$
$-9$

Show the answer

$3^{-2} = \frac{1}{9}$ , so $\log_3 \frac{1}{9} = -2$ . A logarithm can be negative; its input cannot.

Correct: $-2$ : The table pairs $x = -2$ with $\frac{1}{9}$ .
$2$ : $3^2 = 9$ , not $\frac{1}{9}$ ; the reciprocal needs a negative exponent.
$\frac{1}{9}$ : That is the input of the logarithm, not the exponent that produces it.
$-9$ : $3^{-9}$ is far smaller than $\frac{1}{9}$ ; the exponent is $-2$ .

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

$3$ and $4$
$4$ and $5$
$16$ and $17$
$2$ and $3$

Show the answer

$3^3 = 27$ and $3^4 = 81$ , and $27 < 50 < 81$ . So the exponent that gives $50$ is between $3$ and $4$ .

Correct: $3$ and $4$ : $50$ is between $27 = 3^3$ and $81 = 3^4$ .
$4$ and $5$ : $3^4 = 81$ is already more than $50$ .
$16$ and $17$ : That is $50 \div 3$ , about $16.7$ . A logarithm counts factors of $3$ , it does not divide.
$2$ and $3$ : $3^3 = 27$ is less than $50$ , so the exponent is more than $3$ .

Diagram

A logarithmic scale

On this number line, the labeled ticks

1

,

10

,

100

,

1{,}000

and

10{,}000

are equally spaced. Two values are marked:

50

and

2{,}000

.

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. — LevlPrep original diagram.

4. On the scale, a point sits $2$ labeled steps to the right of $10$ . What value does it show?

1,000
30
100
10,000

Show the answer

Each labeled step multiplies by $10$ : $10 \cdot 10 \cdot 10 = 1{,}000$ .

Correct: 1,000: Two steps from $10$ : times $10$ , then times $10$ again.
30: On a logarithmic scale a step multiplies by $10$ ; it does not add $10$ .
100: That is one step to the right of $10$ .
10,000: That is three steps to the right of $10$ .

5. The common logarithm $\log 50$ counts how many labeled steps $50$ sits to the right of $1$ . Which describes $\log 50$ ?

A number between $1$ and $2$
A number between $10$ and $100$
A number between $0$ and $1$
A number between $2$ and $3$

Show the answer

$50$ lies between $10 = 10^1$ and $100 = 10^2$ , so its exponent of $10$ is between $1$ and $2$ (about $1.7$ ).

Correct: A number between $1$ and $2$ : $10^1 < 50 < 10^2$ , so $1 < \log 50 < 2$ .
A number between $10$ and $100$ : Those are the values around $50$ , not their exponents.
A number between $0$ and $1$ : $\log 10 = 1$ already, and $50$ is past $10$ .
A number between $2$ and $3$ : $\log 100 = 2$ , and $50$ is short of $100$ .

6. What is the exact value of $\log_4 8$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Write both as powers of $2$ : $4 = 2^2$ and $8 = 2^3$ . Then $4^{3/2} = 2^3 = 8$ , so $\log_4 8 = \frac{3}{2}$ .

Answer: $\frac{3}{2}$ ≈ 1.500

7. Which equation says the same thing as $5^3 = 125$ ?

$\log_5 125 = 3$
$\log_3 125 = 5$
$\log_{125} 5 = 3$
$\log_5 3 = 125$

Show the answer

$\log_b c = a$ means $b^a = c$ . The base $5$ stays the base, and the logarithm is the exponent $3$ .

Correct: $\log_5 125 = 3$ : Base $5$ , exponent $3$ , value $125$ .
$\log_3 125 = 5$ : The base of the logarithm is the base of the power, $5$ .
$\log_{125} 5 = 3$ : $125^3$ is huge; $\log_{125} 5 = \frac{1}{3}$ .
$\log_5 3 = 125$ : The logarithm gives the exponent, $3$ , not the value $125$ .

Part 9 · Summary

Summary

A logarithm is an exponent:

\log_b c = a

means

b^a = c

. The common logarithm uses base

10

and the natural logarithm uses base

e

. Find exact values by writing the input as a power of the base, and estimate others between consecutive powers. Logarithms take only positive inputs. A logarithmic scale spaces numbers by their logarithms, so equal steps are equal factors.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

Powers of 33

A logarithmic scale

Summary

What comes next

Connections

Powers of $3$