Evaluate log base p of 48

Problem

(log_p)(48)

Solution

1. Identify the expression as a logarithm with an arbitrary base p and an argument of 48

2. Apply the prime factorization of the number 48 to break it down into smaller components.

48=16×3

48=2×3

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

(log_p)(2×3)=(log_p)(2)+(log_p)(3)

4. Apply the power rule for logarithms, which states (log_b)(xn)=n*(log_b)(x)

(log_p)(2)+(log_p)(3)=4*(log_p)(2)+(log_p)(3)

5. Alternatively, apply the change of base formula if a numerical evaluation in terms of natural or common logarithms is required.

(log_p)(48)=ln(48)/ln(p)

Final Answer

(log_p)(48)=ln(48)/ln(p)

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