TMUA Exponentials and Logarithms

Logarithm laws

• $\log_a x+\log_a y=\log_a(xy)$ and $\log_a x-\log_a y=\log_a\dfrac xy$.
• $\log_a x^k=k\log_a x$, $\log_a a=1$ and $\log_a1=0$.
• $\log_a b=c$ means $a^c=b$.

Solving $a^x=b$

Take logs, or write both sides as powers of the same base. When the equation has a mix of powers, look for a substitution such as $u=2^x$.

Check validity

You can only take the log of a positive number, so reject any solution that makes a log argument zero or negative.

Special cases and graphs

Know that $\log_a a^k=k$, so $\log_2 32=5$. Negative and fractional answers are fine: $\log_2\tfrac18=-3$ because $2^{-3}=\tfrac18$. The specification excludes change-of-base questions.

The graph of $y=a^x$ with $a>1$ always passes through $(0,1)$, is always positive and never meets the $x$-axis. The graph of $y=\log_ax$ is its reflection in the line $y=x$.

If a question gives $\log_a2=p$ and asks for $\log_a8$, use the power law: $\log_a8=3\log_a2=3p$. Express every number in the problem through the ones you are given.

Check any answer by substituting it back, especially after a substitution or after squaring, since a spurious solution can appear. A positive base raised to any power stays positive, so $2^x=-4$ has no solution.

Worked examples

Common mistakes

• Writing $\log(x+y)=\log x+\log y$.
• Forgetting to reject a solution that makes a log argument negative.
• Treating $2^{x+1}$ as $2^x+1$.
• Solving $u^2-4u+3=0$ and stopping at $u$ without converting back to $x$.

Test yourself on this topic with our free practice questions.