Math  /  Algebra

QuestionSimplify the following expression. Write as a single logarithm or monomial. 13ln6413ln64=\begin{array}{l} \frac{1}{3} \ln 64 \\ \frac{1}{3} \ln 64= \end{array}

Studdy Solution

STEP 1

1. We are asked to simplify the expression using logarithmic properties.
2. The expression involves a logarithm with a fractional coefficient.
3. The expression can be simplified using the power rule of logarithms.

STEP 2

1. Apply the power rule of logarithms to simplify the expression.

STEP 3

Apply the power rule of logarithms, which states that aln(b)=ln(ba) a \cdot \ln(b) = \ln(b^a) . In this case, a=13 a = \frac{1}{3} and b=64 b = 64 .
13ln64=ln(6413) \frac{1}{3} \ln 64 = \ln(64^{\frac{1}{3}})

STEP 4

Calculate 6413 64^{\frac{1}{3}} . Since 64 is a perfect cube, we have:
6413=4 64^{\frac{1}{3}} = 4
Thus, the expression simplifies to:
ln(4) \ln(4)
The simplified expression is:
ln(4) \boxed{\ln(4)}

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