Exponentials and Logarithms

Exponentials

An exponential function has the variable in the power, e.g. y = aˣ. The special exponential is e^x (e ≈ 2.718), which has the unique property that it is its own derivative: d/dx(eˣ) = eˣ. Exponential models describe growth and decay (e.g. N = N₀e^(kt)).

Logarithms

A logarithm is the inverse of an exponential:

if  y = aˣ   then   x = log_a y

The natural log ln x = log_e x is the inverse of eˣ, so ln(eˣ) = x and e^(ln x) = x.

The laws of logs

log(xy) = log x + log y
log(x/y) = log x − log y
log(xⁿ) = n log x
log_a a = 1        log_a 1 = 0

These let you combine or split logs and, crucially, bring powers down to solve equations.

Solving exponential equations

To solve aˣ = b, take logs of both sides and use the power law:

• 2ˣ = 20 → x log 2 = log 20 → x = log 20 ÷ log 2 ≈ 4.32.
• For equations with e, take ln: e^(2x) = 5 → 2x = ln 5 → x = ½ ln 5.

Linearising with logs (modelling)

For a model y = ax^n, taking logs gives log y = n log x + log a — a straight line (log y against log x): gradient = n, intercept = log a.

For y = ab^x, log y = (log b)x + log a — a straight line of log y against x.

This is how experimental data is used to find the constants of exponential/power models.

Worked example

Solve 3^x = 50, giving x to 3 significant figures.

• Take logs: x log 3 = log 50 → x = log 50 ÷ log 3 = 1.699 ÷ 0.477 = 3.56. ✓

Common mistakes

• Trying to "cross-multiply" instead of taking logs to solve for a power.
• Misusing the laws — log(x + y) is not log x + log y.
• Forgetting ln and eˣ are inverses (they cancel).

Exam tips

• Use the power law to bring the unknown power down, then divide.
• For growth/decay models, take ln of both sides.
• To find model constants, linearise with logs and read gradient/intercept.

Key facts to remember

• eˣ is its own derivative; ln is its inverse (ln eˣ = x).
• Log laws: log(xy) = log x + log y, log(x/y) = log x − log y, log xⁿ = n log x.
• Solve aˣ = b by taking logs; linearise y = axⁿ or y = abˣ with logs to find constants.