Topic 4 of 8

Exponentials and logarithms

Step 13 · Algebra 2 topics

Read logarithms as exponents, switch between exponential and log form, use the log rules, and solve exponential equations with matching bases or with logs.

Logarithms undo exponents

A logarithm answers the question "what power?". $\log_{2} 8 = 3$ because 2³ = 8. In general, $\log_{b} x = y$ means $b^{y} = x$ .

Exponential form	Logarithmic form
2³ = 8	$\log_{2} 8 = 3$
10² = 100	$\log_{10} 100 = 2$
5⁰ = 1	$\log_{5} 1 = 0$
$9^{1/2} = 3$	$\log_{9} 3 = \frac{1}{2}$

Example: Evaluating

Find $\log_{3} 81$ and $\log_{2} \frac{1}{8}$ .

3⁴ = 81, so the first is 4.
2⁻³ is $\frac{1}{8}$ , so the second is −3.

Answer4 and −3

y = 2ˣ and y = log₂ x are mirror images in the dashed line y = x: a log undoes an exponent.

Log rules

Because logs are exponents, the exponent rules become log rules.

Rule	Example
$\log_{b}(MN) = \log_{b} M + \log_{b} N$	$\log_{2} 32 = \log_{2} 4 + \log_{2} 8$
$\log_{b}\frac{M}{N} = \log_{b} M - \log_{b} N$	$\log_{3} 9 = \log_{3} 27 - \log_{3} 3$
$\log_{b}(M^{k}) = k\log_{b} M$	$\log 1000 = 3\log 10 = 3$

Example: Expanding

Expand $\log_{5}\frac{x^{3}}{y}$ .

Use the quotient rule, then the power rule.

Answer $3\log_{5} x - \log_{5} y$

Example: Condensing

Write $2\log x + \log 5$ as a single log.

The power rule turns 2 log x into log x².
The product rule joins the two logs.

Answer $\log(5x^{2})$

Solving exponential equations

To solve an exponential equation, write both sides with the same base if you can. If not, take the log of both sides.

Example: The same base

Solve 4ˣ = 32.

4 = 2² and 32 = 2⁵, so 2²ˣ = 2⁵.
2x = 5.

Answerx = 2.5

Example: Using logs

Solve 3ˣ = 20, to two decimal places.

Take logs of both sides: x log 3 = log 20.
$x = \frac{\log 20}{\log 3}$ , about 1.301 ÷ 0.477.

Answerx ≈ 2.73

Example: Doubling time

$500 grows at 6% a year. When does it reach $1,000?

500 × 1.06ᵗ = 1,000, so 1.06ᵗ = 2.
$t = \frac{\log 2}{\log 1.06}$ , which is about 11.9.

AnswerAfter about 11.9 years

Try it yourself

Work each one on paper first, then open the answer to check.

What is $\log_{4} 64$ ?
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3
What is $\log_{10} 0.01$ ?
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−2
Write 7² = 49 in log form.
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$\log_{7} 49 = 2$
Expand $\log(10x)$ .
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$1 + \log x$
Solve $\log_{2} x = 5$ .
Show answer
x = 32
Solve 9ˣ = 27.
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x = 1.5
Solve 2ˣ = 10, to two decimal places.
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x ≈ 3.32