Expanding logarithms

The examples below will teach you about expanding logarithms using the properties of logarithms.

Expanding logarithms is the inverse of simplifying logarithms. You are trying to express a single logarithm into many individual logarithms.

Example #1

Expand log5 (3xy)

Using the product property, log5 (3xy) = log5 3 + log5 x + log5 y

Example #2

Expand log2 (5x / 10y)

Using 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) 

Example #3

Expand log (a2b3)

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

log10 (a2b3)    = log10 a2 + log10 b3

                       = 2log10 a + 2log10


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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