Simplify/Condense log base 4 of x^2-1-5 log base 4 of x+1

Problem

(log_4)(x2−1)−5*(log_4)(x+1)

Solution

1. Apply the power property of logarithms to the second term, which states n*(log_b)(a)=(log_b)(an)

(log_4)(x2−1)−(log_4)((x+1)5)

2. Apply the quotient property of logarithms, which states (log_b)(M)−(log_b)(N)=(log_b)(M/N)

(log_4)((x2−1)/((x+1)5))

3. Factor the numerator using the difference of squares formula a2−b2=(a−b)*(a+b)

(log_4)(((x−1)*(x+1))/((x+1)5))

4. Simplify the fraction by canceling the common factor of (x+1) from the numerator and denominator.

(log_4)((x−1)/((x+1)4))

Final Answer

(log_4)(x2−1)−5*(log_4)(x+1)=(log_4)((x−1)/((x+1)4))

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