Exponentials and Logarithms Topic Review summary

Key learning points

• Laws: $\log(ab)=\log a+\log b$, $\log(a/b)=\log a-\log b$, $\log(a^k)=k\log a$.
• Change of base: $\log_b a=\frac{\log_k a}{\log_k b}$.
• $a^x=b\Leftrightarrow x=\log_a b$; take logs to solve exponents.
• Natural log $\ln$ pairs with $e^x$.
• Sketch $y=a^x$ and $y=\log_a x$ with intercepts and asymptotes.

Worked example 1 — Solve

Solve $3^{2x}=12$.

Working

Take logs of both sides.

Worked example 2 — Laws

Simplify $\log_2 24-\log_2 3$.

Working

$\log_2 8$.

Worked example 3 — Laws: \\log(ab)=\\log a+\\log b, \\log(a/b

Laws: $\\log(ab)=\\log a+\\log b$, $\\log(a/b)=\\log a-\\log b$, $\\log(a^k)=k\\log a$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Change of base: \\log_b a=\\\log_k a{\\log

Change of base: $\\log_b a=\\frac{\\log_k a}{\\log_k b}$.

Working

Attempt without notes, then check the answers PDF.

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.