Expand the Logarithmic Expression log base 5 of x^2-25

Problem

(log_5)(x2−25)

Solution

1. Identify the expression inside the logarithm as a difference of squares, which follows the pattern a2−b2=(a−b)*(a+b)

2. Factor the quadratic expression x2−25 into (x−5)*(x+5)

3. Apply the product rule for logarithms, which states (log_b)(M*N)=(log_b)(M)+(log_b)(N)

4. Rewrite the expression as the sum of two separate logarithms.

Final Answer

(log_5)(x2−25)=(log_5)(x−5)+(log_5)(x+5)

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