Exponential & Logarithmic Functions

Introduction to Logarithms

Defining a logarithm as the inverse of an exponential function.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Exponential Growth & Decay

The Operation That Undoes an Exponent

Solving 2^x = 8 is easy by inspection: x = 3. But before reading on, think about 2^x = 10 — there's no whole number that works, and no algebra technique from earlier in this course isolates a variable stuck inside an exponent. Every operation in this course so far has had an 'undo': subtraction undoes addition, division undoes multiplication, square roots undo squaring. What operation do you think could undo exponentiation?

Definition — Logarithm

The logarithm base b of x, written log_b(x), is the exponent that b must be raised to in order to produce x: log_b(x) = y means bʸ = x. A logarithm is the inverse function of exponentiation with that same base.

A logarithm is genuinely an inverse function, in exactly the sense you'd expect: an inverse function swaps the roles of input and output, undoing whatever the original function did. Exponentiation takes an exponent and produces a value; a logarithm takes that value and returns the exponent that produced it. log_2(8) = 3 is really just asking 'what power of 2 gives 8,' the same question 2³ = 8 answers in the other direction.

Worked Example — Converting Between Exponential and Logarithmic Form

Rewrite 5³ = 125 in logarithmic form, and log_4(16) = 2 in exponential form. Exponential to log: 5³ = 125 becomes log_5(125) = 3. Log to exponential: log_4(16) = 2 becomes 4² = 16.

Worked Example — Evaluating a Logarithm by Inspection

Evaluate log_3(81). Ask: what power of 3 gives 81? Since 3⁴ = 81, log_3(81) = 4.

Worked Example — Evaluating a Logarithm of a Fractional Value

Evaluate log_2(1/8). Ask: what power of 2 gives 1/8? Since 2⁻³ = 1/8, log_2(1/8) = −3.

Function Explorer

f(x) =

sin(x)

Transform: g(x) = a·f(b(x − h)) + k

a — vertical stretch/reflect1

b — horizontal stretch/reflect1

h — horizontal shift0

k — vertical shift0

Composition

Compose with another function

Trace mode

Show inverse view

Analysis (of the transformed function, in view)

y-intercept: (0, 0)
x-intercepts: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Extrema: local min at (-7.85, -1); local max at (-4.71, 1); local min at (-1.57, -1); local max at (1.57, 1); local min at (4.71, -1); local max at (7.85, 1)
Inflection points: (-9.42, 0), (-6.28, 0), (-3.14, 0), (0, 0), (3.14, 0), (6.28, 0), (9.42, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-1, 1]

Tip

Whenever a logarithm expression feels confusing, rewrite it in exponential form first — log_b(x) = y is often much easier to reason about once it's back in the form bʸ = x.

Common Mistakes

Confusing which number in a logarithm expression is the base and which is the argument, such as evaluating log_3(81) as if 81 were the base.
In log_b(x), the small subscript b is always the base, and x is the value the logarithm is being taken of — write the exponential form, bʸ = x, to keep the roles straight.
Assuming a logarithm can be evaluated for a negative number or zero.
Since a positive base raised to any real exponent always produces a positive result, a logarithm's argument must be positive — log_b of a negative number or 0 is undefined.

Key Takeaways

A logarithm answers 'what exponent produces this value,' making it the inverse function of exponentiation.
log_b(x) = y is equivalent to bʸ = x — the same relationship, written two different ways.
A logarithm's argument must always be positive, since a positive base can never produce a non-positive result.

Summary

Logarithms complete the undo-the-operation pattern from throughout this curriculum, this time undoing exponentiation. The next lesson develops rules for logarithms that mirror the exponent rules they came from.

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