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.