Contract the expressions That is use the properties of logar

Contract the expressions. That is, use the properties of logarithms to write each expression as a single logarithm with a coefficient of 1. Simplify all algebra, if possible. 1/2 log(5x + 1) - [log(x + 1) - log(5x + 1)]

Solution

a) (1/2)log(5x+1) - [ log(x+1) -(3/2)log(5x+1) ]

Use the log properties : logA - logB = log(A/B) ; logA +logB = log(A*B) ; xlogA = logA^x

So, log(5x+1)^1/2 - log(x+1) +log(5x+1)^3/2

log(5x+1)^2 - log(x+1)

log(5x+1)^2/(x+1)

b) ln(x^2 - 81) - ln(x+9)

ln(x^2 -81/(x+9)

ln(x+9)(x-9)/(x+9)

ln(x -9)