Exponential and Logarithmic Functions

A logarithm is an exponent. Every rule in this topic follows from that one sentence, so if a law ever looks arbitrary, translate it back into index form and it becomes obvious.

Laws of indices

$$a^m \times a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}$$

$$a^0 = 1 \qquad a^{-n} = \frac{1}{a^n} \qquad a^{\frac1n} = \sqrt[n]{a} \qquad a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

$$\frac{2^5 \times 2^3}{2^6} = \frac{2^8}{2^6} = 2^2 = 4$$

The exponential function

$y = a^x$ with $a > 0$ and $a \ne 1$ is an exponential function. Every one of them:

• passes through $(0, 1)$, since $a^0 = 1$;
• is positive for all $x$, so the curve never touches the $x$-axis;
• has the $x$-axis as a horizontal asymptote.

For $a > 1$ the curve grows; for $0 < a < 1$ it decays.

The natural exponential uses the constant $e \approx 2.71828$. It matters because $y = e^x$ is the one exponential curve whose gradient at every point equals its own height — a fact that makes calculus far tidier later.

Logarithms

The logarithm is the inverse of the exponential:

$$y = a^x \iff x = \log_a y$$

"$\log_a b$" asks: to what power must $a$ be raised to give $b$?

$$\log_2 32 = 5 \quad \text{because } 2^5 = 32$$ $$\log_3 81 = 4 \quad \text{because } 3^4 = 81$$

Two immediate consequences:

$$\log_a 1 = 0 \qquad \log_a a = 1$$

$\log_{10}$ is written $\log$; $\log_e$ is written $\ln$ and called the natural logarithm.

Laws of logarithms

$$\log_a(xy) = \log_a x + \log_a y$$ $$\log_a\!\left(\frac{x}{y}\right) = \log_a x - \log_a y$$ $$\log_a(x^n) = n\log_a x$$

Each mirrors an index law: multiplying numbers adds their exponents, so it adds their logs.

$$\log 20 - \log 4 = \log\frac{20}{4} = \log 5$$ $$3\log x = \log x^3$$

Change of base

$$\log_a b = \frac{\log_c b}{\log_c a}$$

for any valid base $c$ — usually 10 or $e$, since those are the calculator buttons.

$$\log_2 10 = \frac{\ln 10}{\ln 2} = \frac{2.3026}{0.6931} = 3.32 \text{ (3 s.f.)}$$

Check: $2^{3.32} \approx 10$ ✓

Solving exponential and logarithmic equations

When the unknown is in the exponent, take logs of both sides.

Solve $2^x = 20$.

$$x\log 2 = \log 20$$ $$x = \frac{\log 20}{\log 2} = 4.32 \text{ (3 s.f.)}$$

Solve $3^{2x} = 50$.

$$2x\ln 3 = \ln 50$$ $$x = \frac{\ln 50}{2\ln 3} = 1.78 \text{ (3 s.f.)}$$

When the unknown is inside a logarithm, combine into a single log and undo it.

Solve $\log_2 x + \log_2(x-2) = 3$.

$$\log_2\big(x(x-2)\big) = 3$$ $$x(x-2) = 2^3 = 8$$ $$x^2 - 2x - 8 = 0$$ $$(x-4)(x+2) = 0$$

So $x = 4$ or $x = -2$. But $x = -2$ would require $\log_2(-2)$, which does not exist — reject it.

$$x = 4$$

Check: $\log_2 4 + \log_2 2 = 2 + 1 = 3$ ✓

Reducing to linear form

Experimental data often follows a power law or an exponential law. Taking logs turns either into a straight line, whose gradient and intercept you can measure.

Power law $y = ax^n$:

$$\log y = \log a + n\log x$$

Plot $\log y$ against $\log x$: the gradient is $n$ and the intercept is $\log a$.

Exponential law $y = ab^x$:

$$\log y = \log a + x\log b$$

Plot $\log y$ against $x$ — note, $x$ itself, not $\log x$: the gradient is $\log b$ and the intercept is $\log a$.