Unit 2: Exponential & Logarithmic Functions

Logarithms as Inverses

Relating logarithmic functions to their exponential inverses.

Advanced25 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Exponential Data Modeling

Undoing an Exponential — and Straightening Its Graph

Every exponential model built so far answered 'what output does this input produce?' Before reading on: what tool would you need to answer the reverse question — what input produces a given output — when that input isn't a whole number you can just guess?

Definition — Logarithm as Inverse

The logarithm base b is the inverse of the exponential function base b: logb(y) = x exactly when bˣ = y. Applying a base-b exponential and then a base-b logarithm (in either order) returns the original input, the same inverse relationship every invertible function pair satisfies.

\log_b(y) = x \iff b^x = y

Worked Example — Solving an Exponential Equation with Logarithms

Solve 3ˣ = 20. Taking log base 3 of both sides gives x = log₃(20), which by the change-of-base relationship equals log(20)/log(3) ≈ 1.30103/0.47712 ≈ 2.727. Check: 3^2.727 ≈ e^(2.727 × ln 3) = e^(2.727 × 1.0986) ≈ e^2.996 ≈ 20.0, confirming the solution.

Logarithms don't just undo exponentials — they also straighten exponential data into a line. Applying log to both sides of f(x) = a · bˣ gives log(f(x)) = log(a) + x·log(b), using the product-to-sum property of logarithms. The right side is linear in x, with slope log(b) and intercept log(a). Plotting log(output) against x — a semi-log plot — turns curved exponential growth into a straight line, and the line's slope reveals the base.

\log(a \cdot b^x) = \log a + x \log b

Worked Example — Linearizing Exponential Data with a Semi-Log Plot

Recall f(x) = 5 · 3ˣ from the previous lesson, with outputs 5, 15, 45, 135 at x = 0, 1, 2, 3. Taking log₁₀ of each output: log(5) ≈ 0.699, log(15) ≈ 1.176, log(45) ≈ 1.653, log(135) ≈ 2.130. The differences between consecutive values are 1.176 − 0.699 = 0.477, 1.653 − 1.176 = 0.477, and 2.130 − 1.653 = 0.477 — constant, and equal to log(3), the base's logarithm. The constant difference confirms log(f(x)) is a linear function of x with slope log(3), exactly as the derivation predicts.

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

A constant-ratio table (Unit 2's exponential signature) and a constant-difference semi-log table are the same fact seen two ways — logging an exponential's outputs converts its constant ratios directly into constant differences.

Common Mistakes

Writing log(a · bˣ) as log(a) · log(bˣ), turning a sum into a product.
log(a · bˣ) = log(a) + log(bˣ) = log(a) + x·log(b) — the product-to-sum property converts multiplication inside the log into addition outside it, never leaves a product.
Solving bˣ = y by dividing both sides by b instead of taking a logarithm.
Division only isolates a coefficient, not an exponent — undoing an exponential requires its inverse function, the logarithm, applied to both sides.

Key Takeaways

logb(y) = x is defined so that bˣ = y — the logarithm is the exponential function's inverse.
Taking the logarithm of both sides of an exponential equation isolates a variable exponent that ordinary algebra can't reach.
Because log(a · bˣ) = log(a) + x·log(b), plotting log(output) against x turns exponential data into a line with slope log(b).

Summary

Logarithms undo unrestricted exponential growth. The final lesson of this unit studies a model that builds in its own limit — growth that starts exponential but bends and levels off as it approaches a maximum.

Track your progress

← Exponential Data Modeling Back to Unit 2: Exponential & Logarithmic Functions The Logistic Model→