Expand the Logarithmic Expression log base 6 of cube root of 36

Problem

(log_6)(√(3,36))

Solution

1. Rewrite the radical expression using a fractional exponent.

√(3,36)=36(1/3)

2. Apply the power rule for logarithms, which states (log_b)(xp)=p*(log_b)(x)

(log_6)(36(1/3))=1/3*(log_6)(36)

3. Identify that the base of the logarithm and the argument are related, since 36=6

1/3*(log_6)(6)

4. Apply the power rule again to the inner term.

1/3⋅2*(log_6)(6)

5. Simplify using the property (log_b)(b)=1

2/3⋅1=2/3

Final Answer

(log_6)(√(3,36))=2/3

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