Evaluate natural log of 1/( cube root of e^2)

Problem

ln(1/√(3,e2))

Solution

1. Rewrite the cube root using a fractional exponent.

√(3,e2)=e(2/3)

2. Apply the negative exponent rule to move the expression from the denominator to the numerator.

1/(e(2/3))=e(−2/3)

3. Substitute this back into the natural logarithm expression.

ln(e(−2/3))

4. Use the logarithm power rule ln(ex)=x to simplify the expression.

−2/3

Final Answer

ln(1/√(3,e2))=−2/3

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