Logarithms


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:

\log_a(b)=c.

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_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.

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

a^c=b.

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:

Past Log Exam Questions

LogsExamQuestions

Open LOGS exam questions in New Window

Solomon Basic Log Questions

SolomonBasicLogQuestions

Open Solomon Log questions in New Window

Open Solomon Log solutions in New Window