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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.