# Expanding logarithms

The examples below will teach you about expanding logarithms using the properties of logarithms. First study the example in the figure below carefully so that you can understand the process clearly.

Notice that after expanding logb[(x2y) / z] we get:

logb[(x2y) / z] = 2logb x + logb y - logb z

Notice also that expanding logarithms is the inverse of simplifying logarithms. You are trying to express a single logarithm into many individual logarithms. ## More examples showing how expanding logarithms work

Example #1

Expand log5 (3xy)

In this example, you just need to use the product property.

log5 (3xy) = log5 3 + log5 x + log5 y

Example #2

Expand log (a2b3)

In this example, you need to use a combination of the product property and the power property,

log10 (a2b3)    = log10 a2 + log10 b3

= 2log10 a + 2log10

Example #3

Expand log2 (5x / 10y)

In this example you need to use a combination of the product property and the quotient property,

log2 (5x / 10y)  = log2 (5x) -  log2 (10y)

= log2 (5) +  log2 (x)  -  [ log2 (10) +  log2 (y) ]

= log2 (5) +  log2 (x)  -   log2 (10) -  log2 (y)

=  log2 (5) +  log2 (x)  -   log2 (5×2) -  log2 (y)

=  log2 (5) +  log2 (x)  -   (log2 5 + log2 2) -  log2 (y)

=  log2 (5) +  log2 (x)  -  log2 5 - log2 2 -  log2 (y)

=  log2 (x)  - log2 2 -  log2 (y)

Notice that  log2 (5) -  log2 5 = 0

Notice also that if you remember to simplify a quotient, there will be less work to do as demonstrated below for example #3

log2 (5x / 10y)  =  log2 (x / 2y)

=  log2 x - log2 2y

=  log2 x - ( log2 2 + log2 y )

= log2 x - log2 2 - log2 y

Example #4

Expand log3 [ √ (36x3y) ]

Using a combination of the product property and the power property,

log3 [ √ (36x3y) ]   =  log3 [ (36x3y)1/2 ]

= 1/2 log3 (36x3y)

= 1/2 [ log3 (36) +  log3 (x3) + log3 (y) ]

= 1/2 [ log3 (36) +  3log3 (x) + log3 (y) ]

= 1/2 log3 (36) +  3/2 log3 (x) + 1/2 log3 (y)

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