Exponentials & Logarithms (IB Maths AA SL): notes, exam questions and worked examples

Exponentials & Logarithms notes

Laws of indices and exponential equations

Multiply powers of the same base by adding indices, divide by subtracting, and raise a power to a power by multiplying: $a^m{a}^n=a^{m+n}$, $\frac{a^m}{a^n}=a^{m-n}$, $(a^m{)}^n=a^{mn}$. A negative index is a reciprocal and a fractional index is a root, so $a^{-1/2}=\frac{1}{\sqrt{a}}$.

If both sides can be written with the same base, equate the powers. For example $4^x=8$ becomes $2^{2x}=2^3$, so $x=\frac{3}{2}$.

What a logarithm is

${\log }_{a}b=x$ and $a^x=b$ say the same thing (with $a>0$, $a\ne 1$ and $b>0$). So ${\log }_{2}32=5$ because $2^5=32$. The natural log $\ln x$ is ${\log }_{e}x$, the inverse of $e^x$: $\ln e^x=x$ and $e^{\ln x}=x$.

Two values to know without the booklet: ${\log }_{a}1=0$ and ${\log }_{a}a=1$. You cannot take the log of zero or of a negative number.

The laws of logarithms

Product: ${\log }_{a}xy={\log }_{a}x+{\log }_{a}y$. Quotient: ${\log }_a\frac{x}{y}={\log }_{a}x-{\log }_{a}y$. Power: ${\log }_a{x}^m=m{\log }_{a}x$. Use them to combine several logs into one, for example ${\log }_{3}54-{\log }_{3}2={\log }_{3}27=3$.

The classic slip is to split a log of a sum: $\log (x+y)$ is not $\log x+\log y$.

Change of base

${\log }_{b}a=\frac{{\log }_{c}a}{{\log }_{c}b}$ for any base $c$. It lets you rewrite logs in one base before combining them: ${\log }_{4}x=\frac{{\log }_{2}x}{{\log }_{2}4}=\frac{{\log }_{2}x}{2}$.

Solving exponential and log equations

Unknown in the power: take logs of both sides. $3^x=20$ gives $x=\frac{\ln 20}{\ln 3}\approx 2.73$.

Logs on both sides: combine into a single log, then undo it. ${\log }_{2}x+{\log }_2(x-2)=3$ gives $x(x-2)=8$, so $x=4$ or $x=-2$. Reject $x=-2$: it makes the arguments negative.

Hidden quadratics: $e^{2x}-5e^x+6=0$ is a quadratic in $u=e^x$: $u=2$ or $u=3$, so $x=\ln 2$ or $x=\ln 3$. On Paper 1 leave answers exact like this.