Exponential Growth and Decay

Exponential functions: growth vs decay

$y = b^x$ behaves fundamentally differently depending on the base $b$:

• $b > 1$ (growth): The function increases rapidly. Example: $y = 2^x$ doubles every time $x$ increases by 1. Population growth, compound interest.
• $0 < b < 1$ (decay): The function decreases toward zero. Example: $y = (1/2)^x$ halves every time $x$ increases by 1. Radioactive decay, depreciation.

Key properties (both cases)

• The y-intercept is always $(0, 1)$ because $b^0 = 1$.
• The x-axis ($y = 0$) is a horizontal asymptote — the curve approaches but never touches it.
• The function is always positive ($b^x > 0$ for all $x$).

Growth vs decay comparison

At $x = 10$: $2^{10} = 1024$ (explosive growth) vs $(1/2)^{10} = 1/1024 \approx 0.001$ (nearly zero).

The doubling time (for growth) is $t = \ln 2 / \ln b$. The half-life (for decay) is $t = \ln 2 / \ln(1/b)$.

Try it in the visualization

Adjust the base $b$. Watch the curve switch from growth ($b > 1$) to decay ($b < 1$). Both curves share the point $(0, 1)$. The doubling time or half-life is shown. The asymptote at $y = 0$ is always present. Growth doubles every fixed interval; decay halves. The base $b$ controls the rate.