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.