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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