Exponential and Logarithmic Functions: Concepts and Examples

An exponential function has the form f(x) = a^x with a > 0, a ≠ 1, and the variable in the exponent. The logarithmic function g(x) = log_a x is its inverse: it answers "to what power must I raise a to get x?" — so log_2 8 = 3 because 2^3 = 8.

These two functions matter because they describe growth and decay that multiply rather than add. Populations, compound interest, radioactive decay, sound loudness and earthquake magnitudes all follow exponential or logarithmic laws. If you can read these two graphs and flip between the exponential and logarithmic forms, you can solve any equation where the unknown is trapped in an exponent.

Key Concepts

• Exponential function: f(x) = a^x with a > 0, a ≠ 1. Its domain is all real numbers and its range is (0, ∞) — it never touches zero or goes negative.
• Natural base e: e ≈ 2.71828; the natural log ln x = log_e x appears in continuous growth formulas.
• Inverse relationship: a^log_a x = x for x > 0, and log_a (a^x) = x for all real x.

Formulas

• Log laws: log_a(mn) = log_a m + log_a n; log_a(m/n) = log_a m − log_a n; log_a(m^r) = r·log_a m. Valid for m, n > 0, a > 0, a ≠ 1.
• Exponential growth: P(t) = P_0·a^t, where P_0 is the starting amount and a is the growth factor per unit time. Assumption: the growth rate stays constant.
• Solving exponential equations: if a^x = b with a > 0, a ≠ 1, b > 0, then x = log_a b = ln b / ln a.

Worked Example

A culture starts with 500 bacteria and doubles every 3 hours. How many are there after 12 hours, and when does it reach 8000?

1. Model: P(t) = 500 · 2^t/3, where t is in hours.
2. After 12 hours: t/3 = 4, so P = 500 · 2^4 = 500 · 16 = 8000 bacteria.
3. Check the arithmetic: 2^4 = 16, and 500 × 16 = 8000. Correct.
4. Time to reach 8000 (verify by logs): 500 · 2^t/3 = 8000 gives 2^t/3 = 16 = 2^4, so t/3 = 4 and t = 12 hours, matching the answer above.
5. A non-exact case: to reach 6000, solve 2^t/3 = 12, so t/3 = log_2 12 = ln 12 / ln 2 ≈ 2.4849/0.6931 ≈ 3.585, giving t ≈ 10.75 hours.

Common Misconceptions

• "log(a + b) = log a + log b." False — that rule is for multiplication: log(ab) = log a + log b. There is no simplification for log(a + b).
• "a^x can be negative or zero." No. For a > 0, a^x is always positive, which is why log_a x is only defined for x > 0.
• "Exponential growth means adding the same amount each time." That is linear growth. Exponential means multiplying by the same factor each period — the additions get bigger and bigger.

Real-World Applications

• Compound interest: A = P(1 + r)^t is exponential in time t; small rate differences compound into large differences over decades.
• Radioactive decay: remaining mass follows N = N_0(1/2)^t/T with half-life T, used in carbon dating.
• pH and decibels: pH = −log_10[H^+] and sound level in dB = 10·log_10(I/I_0) compress huge ranges into small numbers.

Exponential functions grow by multiplication and never reach zero, while logarithms undo them and are defined only for positive inputs. The log laws turn multiplication into addition — the key trick for solving exponential equations and measuring pH, decibels and earthquake strength.