# Logs – Bases

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.

You can change the base of the log using the following formula:

$\log_a(x)=\frac{\log_b(x)}{\log_b(a)}$

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.

Example 2

Solve the equation   $\log_2(x)=\log_x(5)$.

Notice that the bases are different and so we must make them the same first using the change of base formula. Choose to change the right hand side so that 2 is the base for all terms:

$\log_x(5)=\frac{\log_2(5)}{\log_2(x)}$

and substituting into the equation gives

$\log_2(x)=\frac{\log_2(5)}{\log_2(x)}$

Multiplying both sides by $\log_2(x)$ gives

$\log_2(x)^2=\log_2(5)=2.322$

Note that there is no log rule for multiplying logs, only for multiplying within the log and also note that $\log_2(5)$ can be calculated using the calculator.

$\log_2(x)=1.524$

This can be rewritten as

$x=2^{1.524}$ (see below*)

and so x=2.875 to 3 decimal places.

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

See more on log rules.