Logarithms

Starting point: Years 9 and 10

What a logarithm is

A logarithm answers the question: what power do I need?

$${\log }_{a}y=x\text{means}{a}^x=y$$

${\log }_{2}8=3$ because $2^3=8$. Read it as "log base 2 of 8 is 3."

The log laws

These come straight from the index laws.

Solving with logs

When you cannot match bases, take logs of both sides.

$$2^x=10\Rightarrow x={\log }_{2}10=\frac{{\log }_{10}10}{{\log }_{10}2}\approx 3.32$$

Quick summary

• ${\log }_{a}y=x$ means $a^x=y$. A log is a power.
• Product: add logs. Quotient: subtract logs. Power: bring the power to the front.
• ${\log }_{a}1=0$ and ${\log }_{a}a=1$.
• $\ln x={\log }_{e}x$ undoes $e^x$. Change of base: ${\log }_{a}x=\frac{\ln x}{\ln a}$.
• $y={\log }_{a}x$ passes through $(1,0)$ with asymptote $x=0$.

Quiz