# Logarithms

Logarithms, or logs for short, provide a way to extract unknown powers in an expression. When you come across logs, you will see the word ‘log’ followed by a small number that we call a subscript – this subscript is known as the ‘base’. This is followed by a number in brackets although sometimes the brackets are left out.

$\log_a(b)=c$

When reading the expressions, replace ‘log’ with ‘the power of’ and this reads as ‘the power of a to get a result of b is c’. So it is evident that this can be used interchangeably with the equation

$a^c=b$

Example$\log_2(8)=3$ can be changed to $2^3=x$ and so x is 8.

Some students find logarithms or logs quite confusing. It can be helpful to read logs in a certain way. For instance, $\log_a(b)$ can be thought of as the power of a that gives b. a is known as the base of the logarithm.

For example, in $\log_2(8)$ the base is 2 and since it can be read as the power of 2 to give 8, the value of $\log_2(8)$ is 3.

Some other examples include:
$\log_3(9)=2$
$\log_4(64)=3$
$\log_2(1/8)=-3$ since $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$.
$\log_{123}(1)=0$ since anything to the power of 0 is 1.

See more on logs:

*Quite often, in exam questions where logs are used, it is useful to use $\log_a(b)=c\$ and $a^c=b$ interchangeably.

Example 1

There is a button on your calculator that can help you with powers that are not calculable in your head. The button has the word log followed by a two boxes; insert your base into the small lower box and the number that you see in brackets in the second. Use it to verify the following:

a)  $\log_{9}(5)=0.732$ to 3 decimal places.

b)  $\log_{8}(9)=1.06$ to 2 decimal places.

c)  $\log_{12}(150)=2.016$ to 3 decimal places.

d)  $\log_{2}(0.7)=-0.514$ to 3 decimal places.

e)  $\log_{3}(-4)=N/A$

Note that the final example has no solution – it is not possible to determine the power of 3 that gives a result of -4. There is no amount of times you can multiply 3 by itself to get any negative number. Your calculator will verify this with a MATH ERROR.

See more on log rules.