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 3
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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