Expanding logarithms

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

After expanding log_b[(x²y) / z] we get:

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

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

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)