Use the Laws of Logarithms to rewrite the expression

Question

Use the Laws of logarithms to rewrite the expression.

ln((x^5sqrtx-1)/(3x-12))

in a form with no logarithm of a product, quotient, or power. After rewriting we have

ln((x^5sqrtx-1)/3x-12))= Alnx+Bln(x-1)+Cln(3x-12)

with the constant A=

the constant B=

and the constant C=

Answer 1

This is of the form of Ln(a/b) = Ln(a) - Ln(b), Ln( x⁵√x-1) / (3x-12)) where a = x⁵√x-1 , and b = (3x-12)

Ln( x⁵√x-1) / (3x-12)) = Ln( x⁵√x-1) - Ln(3x-12) , Now the term Ln( x⁵√x-1) is of the form Ln(ab) = Ln(a)+Ln(b)

˜

Here a = x⁵ , and b = √x-1, then Ln( x⁵√x-1) = Ln(x⁵) + Ln√x-1 and the expression can be written as:

Ln(x⁵) + Ln√x-1 - Ln(3x-12) , or Ln(x⁵) + Ln(x-1)^1/2 - Ln(3x-12) , now we know from logarithmic rules that

Ln(x)^a = a Ln(x) , lets apply that and we get, 5Ln(x)+1/2ln(x-1) -Ln(3x-12)

The desired form is Alnx+Bln(x-1)+Cln(3x-12), comparing coefficients,

A = 5, B =1/2, and C=-1