Logarithmic Function — Definition, Formula & Examples

A logarithmic function is the inverse of an exponential function — it answers the question "what exponent do I need?" For example,

\log_2(8) = 3

because

2^3 = 8

The logarithmic function with base $b$ (where $b > 0$ and $b \neq 1$ ) is defined as $f(x) = \log_b(x)$ , where $\log_b(x) = y$ if and only if $b^y = x$ . Its domain is $(0, \infty)$ and its range is $(-\infty, \infty)$ .

Key Formula

y = \log_b(x) \quad \Longleftrightarrow \quad b^{\,y} = x

Where:

$b$ = Base of the logarithm (b > 0, b ≠ 1)
$x$ = Argument (input); must be positive
$y$ = Output — the exponent that b must be raised to in order to equal x

How It Works

To evaluate

\log_b(x)

, ask yourself: "

b

raised to what power gives

x

?" The graph of

y = \log_b(x)

passes through

(1, 0)

for every base, because

b^0 = 1

. When

b > 1

, the function increases slowly and has a vertical asymptote at

x = 0

. When

0 < b < 1

, the function is decreasing instead. You can convert between logarithmic and exponential form freely:

\log_b(x) = y

is the same statement as

b^y = x

Worked Example

Problem: Evaluate

\log_5(125)

Rewrite as an exponential question: Ask: 5 raised to what power equals 125?

5^{\,y} = 125

Find the exponent: Since

5^3 = 125

, the exponent is 3.

5^3 = 125

State the result: Therefore the logarithm equals 3.

\log_5(125) = 3

Answer:

\log_5(125) = 3

Why It Matters

Logarithmic functions appear whenever you solve exponential equations — for instance, finding how long an investment takes to double or how many half-lives a radioactive sample has undergone. They also form the basis of the Richter scale, the decibel scale, and the pH scale in chemistry.

Common Mistakes

Mistake: Trying to take the logarithm of zero or a negative number.

Correction: The domain of

\log_b(x)

x > 0

only. Expressions like

\log_2(-4)

\log_3(0)

are undefined in the real numbers.