Evaluate log base 5 of fifth root of 125

Problem

(log_5)(√(5,125))

Solution

1. Rewrite the radicand 125 as a power of the base 5

125=5

2. Express the fifth root using a fractional exponent.

√(5,5)=(5)1/5

3. Apply the power of a power rule for exponents by multiplying the indices.

(5)1/5=5

4. Substitute the exponential form back into the logarithmic expression.

(log_5)(5)

5. Use the inverse property of logarithms, (log_b)(bx)=x to simplify the expression.

(log_5)(5)=3/5

Final Answer

(log_5)(√(5,125))=3/5

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