Exponentials and Logarithms summary

Key learning points

• $\log_a(xy)=\log_a x+\log_a y$; $\log_a(x^k)=k\log_a x$.
• $a^x=b\Rightarrow x=\log_a b$; change of base $\log_a b=\frac{\ln b}{\ln a}$.
• Graphs of $y=a^x$ and $y=\log_a x$ reflect in $y=x$.
• Domain of $\log$: argument must be positive.

Worked example 1 — Solve

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

Working

Take logs of both sides; any consistent base works.

Worked example 2 — Key skill 1

Use this method from the Exponentials and Logarithms sheet: $\log_a(xy)=\log_a x+\log_a y$; $\log_a(x^k)=k\log_a x$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — change of base

Use this method from the Exponentials and Logarithms sheet: $a^x=b\Rightarrow x=\log_a b$; change of base $\log_a b=\frac{\ln b}{\ln a}$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Graphs of and reflect in

Use this method from the Exponentials and Logarithms sheet: Graphs of $y=a^x$ and $y=\log_a x$ reflect in $y=x$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Domain of : argument must be positive

Use this method from the Exponentials and Logarithms sheet: Domain of $\log$: argument must be positive.

Working

Each key learning point matches a question type on the printable worksheet.

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.