Functions + inverses • visual math studio

See exponentials and logarithms reflect into each other.

Adjust the base, compare y = b^x and y = log_b(x), and watch how the inverse relationship stays locked to the diagonal line y = x. This makes growth, decay, and domain-range swapping feel concrete instead of abstract.

Exponential + logarithm together

Growth and decay bases

Reflection across y = x

Mapped inverse points

Current model

y = 2^x
y = log₂(x)
y = x

BehaviorGrowth, because b > 1.

Inverse ideaPoints swap coordinates across y = x.

Choose the base

Move the base to explore steep growth, gentle growth, or exponential decay. The logarithm updates at the same time because it is the inverse of the exponential.

Base b 2.00

b must be positive and cannot equal 1.

Tracked x-value on y = b^x 1.50

The highlighted inverse point will swap coordinates from (x, b^x) to (b^x, x).

Quick presets

Highlighted exponential point

(1.50, 2.83)

Matching inverse point

(2.83, 1.50)

Domain / range reminder

Exp: all x, y > 0. Log: x > 0, all y.

Teacher cue

When b > 1, the exponential rises and the logarithm rises more slowly.

Live inverse graph

Both functions are drawn on the same axes, along with the mirror line y = x.

y = b^x

y = log_b(x)

y = x

Inverse point pair

Growth vs. decay
If b > 1, both graphs increase. If 0 < b < 1, both graphs decrease. The inverse relationship stays true either way.

Why the graphs mirror
To undo y = b^x, you swap the input and output. That turns (x, b^x) into (b^x, x), which lands on y = log_b(x).