Exponential and logarithmic functions

The core idea

Linear change adds roughly the same amount each step (earn $50 more each month). Exponential change multiplies by roughly the same factor each step (a population that grows 3% per year, or a video that doubles views each day for a short streak). The formula family is:

y = a · bˣ

Here a is a starting value (what you have when x = 0), and b is the growth factor per unit of x. If b > 1, the graph climbs faster and faster. If 0 < b < 1, it decays toward zero. If b = 1, nothing changes—a flat line.

Where you already see this

• Bank interest / savings apps — Compound interest multiplies the balance by a factor each period. “7% per year” is exponential thinking, not a flat $7 added forever.
• Cooling coffee — A hot cup approaches room temperature; the gap shrinks by a fraction of itself over equal time intervals (Newton’s law of cooling is exponential decay in the temperature difference).
• Screen brightness or volume steps — Many devices use scales that feel even to humans but are based on multiplicative steps.
• Viral shares (short term) — If each person tells two new people, counts can explode—until limits kick in. The early phase is exponential-shaped.
• Radioactive half-life (science class) — Each half-life multiplies remaining amount by ½.
• pH and earthquake magnitude — These are logarithmic scales: equal steps on the scale mean multiplying the underlying quantity (acidity, energy), not adding a fixed amount.

So when someone asks “what is this used for?” a fair answer is: anywhere a percentage change or a doubling/halving rule is more honest than a fixed add-on.

Reading the graph

An exponential growth curve looks slow at first if you start small, then steepens. Decay looks steep at first, then flattens toward a floor. That shape is the visual signature. Linear graphs never “bend up” like true exponential growth.

Logarithms: the inverse question

If exponential asks “start at a, multiply by b, x times—what do you get?”, a logarithm asks the reverse: “I ended up with this amount; what exponent x produced it?”

Symbolically, if y = bˣ (with b > 0, b ≠ 1), then x = log_b(y). Logs undo exponentials the way subtraction undoes addition. On a graph, the log curve is the reflection of the exponential across the line y = x.

Everyday log thinking: “How many doublings to go from 1 to 1024?” is a log base 2 question (answer 10). Earthquake magnitudes and pH compress huge ranges of raw values into smaller numbers humans can compare.

Connecting to later math

Calculus will ask how fast exponential quantities change. The surprising fact is that the rate of change of eˣ is again eˣ—one reason base e appears in continuous growth. Precalculus builds fluency with the shapes and inverses first so that fact is interesting instead of opaque.

Related

• Functions and transformations
• Sequences (discrete multiply-by-r cousin)
• Rates and limits