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Math  /  Algebra

QuestionExpress ln⁡543\ln \sqrt[3]{54} in terms of ln⁡2\ln 2 and/or ln⁡3\ln 3. ln⁡543=\ln \sqrt[3]{54}= □\square (Type an exact answer.)

Studdy Solution

STEP 1

1. We are given the expression ln⁡543\ln \sqrt[3]{54}.
2. We need to express this in terms of ln⁡2\ln 2 and ln⁡3\ln 3.
3. We can use properties of logarithms to simplify the expression.

STEP 2

1. Rewrite the expression ln⁡543\ln \sqrt[3]{54} using the property of logarithms for roots.
2. Express 5454 in terms of its prime factors.
3. Use the properties of logarithms to separate the expression into terms involving ln⁡2\ln 2 and ln⁡3\ln 3.

STEP 3

Rewrite the expression using the property of logarithms for roots:
ln⁡543=13ln⁡54\ln \sqrt[3]{54} = \frac{1}{3} \ln 54

STEP 4

Express 5454 in terms of its prime factors:
54=2×3354 = 2 \times 3^3

STEP 5

Use the properties of logarithms to separate the expression:
ln⁡54=ln⁡(2×33)=ln⁡2+ln⁡33\ln 54 = \ln (2 \times 3^3) = \ln 2 + \ln 3^3
Apply the power rule of logarithms:
ln⁡33=3ln⁡3\ln 3^3 = 3 \ln 3
So,
ln⁡54=ln⁡2+3ln⁡3\ln 54 = \ln 2 + 3 \ln 3

STEP 6

Substitute back into the expression from Step 1:
13ln⁡54=13(ln⁡2+3ln⁡3)\frac{1}{3} \ln 54 = \frac{1}{3} (\ln 2 + 3 \ln 3)
Distribute the 13\frac{1}{3}:
=13ln⁡2+ln⁡3= \frac{1}{3} \ln 2 + \ln 3
Thus, the expression ln⁡543\ln \sqrt[3]{54} in terms of ln⁡2\ln 2 and ln⁡3\ln 3 is:
13ln⁡2+ln⁡3\boxed{\frac{1}{3} \ln 2 + \ln 3}

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