Exponential and Logarithmic Functions summary

Key learning points

• $\frac{d}{dx}(e^{kx})=ke^{kx}$; $\int e^{kx}\,dx=\frac{1}{k}e^{kx}+C$.
• $\frac{d}{dx}(\ln|x|)=\frac{1}{x}$.
• Log laws still apply with $\ln$.
• Growth/decay: $x=Ae^{kt}$ with $k$ from data.

Worked example 1 — Evaluate

$f(x)=2x-1$. Find $f(5)$.

Working

Substitute.

Worked example 2 — Composite

$f(x)=x+1$, $g(x)=2x$. Find $f(g(3))$.

Working

Apply $g$ then $f$.

Worked example 3 — Inverse

$f(x)=3x+2$. Find $f^{-1}(x)$.

Working

Swap and rearrange.

Worked example 4 — Chain rule ln

Differentiate $y=\ln(3x^2+1)$.

Working

Derivative of the inside over the inside.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.