# How to simplify logarithms

Find out how to simplify logarithms by writing a logarithmic expression as a single logarithm with these exercises.

## Examples showing how to simplify logarithms

Exercise #1

Simplify log3 40 - log3 10

Using the quotient property, log3 40 - log3 10 = log3  40 / 10

Simplify log3  40 / 10  to get log3 4

log3 40 - log3 10 = log3 4

Exercise #2

Simplify log4 3 + log4 6

Using the product property, log4 3 + log4 6 = log4  3 x 6

Simplify log4  3 x 6  to get log4 18

log4 3 + log4 6 = log4 18

Exercise #3

Simplify log10 9 + log10 5  - log10 15

Using a combination of the product rule and the quotient rule, we can simplify this logarithmic expression  as shown below.

log10 9 + log10 5  - log10 15   =   log10 ( 9 x 5 )  - log10 15

=   log10 ( 45 )  - log10 15

=   log10 ( 45 / 15 )

=   log10

Exercise #4

Simplify log5 1 / 8 + 3 log5 4

Using a combination of the product rule and the power rule, we can simplify as shown below.

log5 1 / 8 + 3 log5 4   =  log5 1 / 8 +  log5 43

=  log5 1 / 8 +  log5 64

=  log5 (1 / 8 x 64 )

=  log5 ( 64 / 8 )

= log5 8

Here is yet another example clearly showing how to simplify a logarithmic expression using the properties of logarithms.

In the example below, we use the power property and the product property to simplify log6 24 + 2 log6 3. Logarithms can be simplified using only one property or a combination of all 3 properties.

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