What is the difference between exponential growth and decay?

Exponential growth occurs when the base a > 1 — the function y = a^x increases rapidly as x increases. Exponential decay occurs when 0 < a < 1 — the function decreases toward zero. For example, 2^x grows (doubles), while (1/2)^x decays (halves).

Why are exponential and logarithmic functions inverses?

They undo each other: if y = a^x, then x = log_a(y). Graphically, they are reflections of each other across the line y = x. Every point (p, q) on the exponential becomes (q, p) on the logarithm.

What is special about base e?

The number e ≈ 2.71828 is special because the derivative of e^x is itself: d/dx(e^x) = e^x. This makes it the natural base for calculus. The natural logarithm ln(x) = log_e(x) appears throughout mathematics, physics, and engineering.

Exponential & Logarithm Explorer

Q: Why are exponential and logarithmic functions inverses?

They undo each other: if y = a^x, then x = log_a(y). Graphically, they are reflections of each other across the line y = x. Every point (p, q) on the exponential becomes (q, p) on the logarithm.

Q: What is special about base e?

The number e ≈ 2.71828 is special because the derivative of e^x is itself: d/dx(e^x) = e^x. This makes it the natural base for calculus. The natural logarithm ln(x) = log_e(x) appears throughout mathematics, physics, and engineering.

Adjust the base and watch exponential growth/decay and its inverse logarithm. They reflect across y = x.

y = a^x & y = log_a(x)

Presets

Base a = 2.72

Show

y = a^x y = log_a(x) y = x line

Properties

Base	--
Exp equation	--
Log equation	--
Exp key pts	--
Log key pts	--
Growth/Decay	--

The Math Behind It

Exponential Functions

Form: y = a^x where a > 0, a ≠ 1
Always passes through (0, 1)
Horizontal asymptote: y = 0
a > 1: exponential growth
0 < a < 1: exponential decay
Special base: e ≈ 2.718 (natural exponential)

Logarithmic Functions

Form: y = log_a(x) — inverse of a^x
Always passes through (1, 0)
Vertical asymptote: x = 0
Domain: x > 0 only
log_a(a) = 1, log_a(1) = 0
ln(x) = log_e(x) — natural logarithm

Inverse Relationship

a^x and log_a(x) are reflections across y = x
If (p, q) is on a^x, then (q, p) is on log_a(x)
a^log_a(x) = x and log_a(a^x) = x

Exponential & Logarithm Explorer

y = ax & y = loga(x)

Properties

The Math Behind It

Exponential Functions

Logarithmic Functions

Inverse Relationship

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y = a^x & y = log_a(x)