# Logs – Bases

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

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

Some other examples include:

since .

since anything to the power of 0 is 1.

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

**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) to 3 decimal places.

b) to 2 decimal places.

c) to 3 decimal places.

d) to 3 decimal places.

e)

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 .*

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:

and substituting into the equation gives

Multiplying both sides by gives

Note that there is no log rule for multiplying logs, only for multiplying within the log and also note that can be calculated using the calculator.

This can be rewritten as

(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 and interchangeably.

See more on log rules.