Logarithms, or logs for short, are essentially powers and are useful when a power is unknown.

When you come across logs, you will usually see the word ‘log’ followed by a small subscript then a number in brackets:


The subscript is known as the base and the number in brackets (although sometimes the brackets are left out) is the exponent.

It can help to understand logs by making a habit of, when reading log expressions, saying ‘the power of’ instead of the word ‘log’. The above reads as ‘the power of a to get a result of b is c’. 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_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.

The equation \log_a(b)=c can be used interchangeably with the equation


E.g. \log_5(25)=x can be changed to 5^x=25 and so x is 2.

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)=-0.515 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 logs: