FoundationsConcept reference

Logarithms

A base-b logarithm of a positive number is the exponent to which a positive base other than one must be raised to obtain that number.

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On this page 7 sections

Overview
Since 2^3=8, the exponent that turns base 2 into 8 is 3
The answer can be zero or negative
Check your understanding
Key takeaway
Sources & further reading
Concept connections

01Since 2^3=8, the exponent that turns base 2 into 8 is 3#

Since $2^3=8$ , the exponent that turns base 2 into 8 is 3. We write $\log_2(8)=3$ , read “log base 2 of 8.”

Figure 1The logarithm graph has y=0 at x=1, y=1 at x=2, y=2 at x=4 and y=3 at x=8. Reading x=8 returns exponent 3, because 2 cubed is 8.

A logarithm asks the exponent question.

\log_b(x)=y

This means $b^y=x$ . Here, $b$ is the base, $x$ the argument, and $y$ the answer. For real logarithms, use $x>0$ , $b>0$ , and $b\ne1$ .

Check your reasoning

What is $\log_3(27)$ ?

Show answer and explanation

Because $3^3=27$ , the required exponent is 3.

02The answer can be zero or negative#

The answer can be zero or negative. Since $2^0=1$ , $\log_2(1)=0$ . Since $2^{-1}=1/2$ , $\log_2(1/2)=-1$ . The arguments stay positive in both cases.

Check your reasoning

What is $\log_2(1/4)$ ?

Show answer and explanation

-2

The exponent is −2 because $2^{-2}=1/4$ .

03Check your understanding#

Check your reasoning

A quantity is multiplied by 3 each round. After how many rounds is it 81 times its starting value?

Show answer and explanation

The total multiplier after n rounds is $3^n$ . Since $3^4=81$ , it takes 4 rounds; this is $\log_3(81)$ .

Key takeaway

Ask which exponent makes the given base produce the argument.

Evaluate a logarithm as an inverse exponent

Sources & further reading

[1]
OpenStax College Algebra 2e: 6 3 Logarithmic Functions ↗OpenStax · Book

01Since 2^3=8, the exponent that turns base 2 into 8 is 3#

02The answer can be zero or negative#

03Check your understanding#

Ask which exponent makes the given base produce the argument.

Sources & further reading

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