Math Log Problem Please help

Question

use the properties of logarithms to expand the logarithmic expression wherever possible evaluate logarithmic expressions

ln((x+4)/(x²-16)⁵)^(4/5)

Answer 1

To expand, we write the single logarithm as a sum or difference of logs. 

 

 

First, we can bring the outer exponent, 4/5, as the coefficient of the log.

 

(4/5)ln[(x + 4) / (x² - 16)⁵]

 

 

Notice that the arguments of the logs are being divided.  This tells us to write a difference of logs.

 

(4/5)ln(x + 4) - (4/5)ln(x² - 16)⁵

 

 

Then we can bring the exponent on the second log term as the coefficient of the log.

 

(4/5)ln(x + 4) - 4ln(x² - 16)

 

 

Notice that the argument on the second term of the log is a difference in perfect squares.  Thus, when factor the arguments using FOIL.  FOIL is multiplying two binomial factors.  Those terms are the product of the arguments.

 

(4/5)ln(x + 4) - [4ln(x + 4) + 4ln(x - 4)] =

 

(4/5)ln(x + 4) - 4ln(x + 4) - 4ln(x - 4) =

 

 

(4/5)ln(x + 4) - (20/5)ln(x + 4) - 4ln(x - 4)

 

 

Combine common log terms to get

 

- (16/5)ln(x + 4) - 4ln(x - 4)

 

 

 

Answer 2

ln{[(x+4)/(x²-16)⁵]}^4/5 = (4/5) ln[(x+4)/(x²-16)⁵] = (4/5) [ln(x+4) - 5 ln(x²-16)]

 

= (4/5) {ln(x+4) - 5 ln[(x-4)(x+4)]} = (4/5) [ln(x+4) - 5 ln(x-4) - 5 ln(x+4)] 

 

= (4/5) [-4 ln(x+4) - 5 ln(x-4)] = -(4/5) [4 ln(x+4) + 5 ln(x-4)]

 

Simplifying as far as possible, you are left with

 

(-16/5) ln(x+4) - 4 ln(x-4)