Find out how to simplify logarithms by writing a logarithmic expression as a single logarithm with these exercises.
Exercise #1
Simplify log_{3} 40 - log_{3} 10
Using the quotient property, log_{3} 40 - log_{3} 10 = log_{3} 40 / 10
Simplify log_{3} 40 / 10 to get log_{3} 4
log_{3} 40 - log_{3} 10 = log_{3} 4
Exercise #2
Simplify log_{4} 3 + log_{4} 6
Using the product property, log_{4} 3 + log_{4} 6 = log_{4} 3 x 6
Simplify log_{4} 3 x 6 to get log_{4} 18
log_{4} 3 + log_{4} 6 = log_{4} 18
Exercise #3
Simplify log_{10} 9 + log_{10} 5 - log_{10} 15
Using a combination of the product rule and the quotient rule, we can simplify this logarithmic expression as shown below.
log_{10} 9 + log_{10} 5 - log_{10} 15 = log_{10} ( 9 x 5 ) - log_{10} 15
= log_{10} ( 45 ) - log_{10} 15
= log_{10} ( 45 / 15 )
= log_{10} 3
Exercise #4
Simplify log_{5} 1 / 8 + 3 log_{5} 4
Using a combination of the product rule and the power rule, we can simplify as shown below.
log_{5} 1 / 8 + 3 log_{5} 4 = log_{5} 1 / 8 + log_{5} 4^{3}
= log_{5} 1 / 8 + log_{5} 64
= log_{5} (1 / 8 x 64 )
= log_{5} ( 64 / 8 )
= log_{5} 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 log_{6} 24 + 2 log_{6} 3. Logarithms can be simplified using only one property or a combination of all 3 properties.
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