Exponential and Logarithmic Functions

Exponential Growth

Evaluate A(t) = 100 · 2^t at t = 1, 2, 3 to observe exponential growth doubling at each step.

Example: A(t) = 100 · 2^t starts at A(0) = 100 and doubles each step: A(1) = 200, A(2) = 400, A(3) = 800.

Example

Evaluate an exponential growth model at successive input values.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

A(t)=100⋅2tA(t)= 100 \cdot 2 ^{t}

Step 2 — Evaluate t = 1

Evaluate at t = 1: 100 times 2^1 = 200.

A(1)=100⋅21=200A( 1 )= 100 \cdot 2 ^{ 1 }= \hl{200}

Step 3 — Evaluate t = 2

Evaluate at t = 2: 100 times 2^2 = 400.

A(2)=100⋅22=400A( 2 )= 100 \cdot 2 ^{ 2 }= \hl{400}

Step 4 — Evaluate t = 3

Evaluate at t = 3: 100 times 2^3 = 800.

A(3)=100⋅23=800A( 3 )= 100 \cdot 2 ^{ 3 }= \hl{800}

Step 5 — Result

The final growth value is 800.

A(3)=800A( 3 )= \hl{800}
exponential-growth An exponential function has the form A(t) = A₀ · bᵗ where A₀ is the initial value and b > 1 is the growth factor. Each unit increase in t multiplies the output by b.