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Math  /  Algebra

QuestionUse properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1 . Evaluate logarithmic expressions if possible. 7ln⁡x−18ln⁡y7ln⁡x−18ln⁡y=□\begin{array}{l} 7 \ln x-\frac{1}{8} \ln y \\ 7 \ln x-\frac{1}{8} \ln y=\square \end{array}

Studdy Solution

STEP 1

1. We are given a logarithmic expression to condense.
2. The expression should be written as a single logarithm with a coefficient of 1.
3. Properties of logarithms such as the power rule, product rule, and quotient rule can be used.

STEP 2

1. Apply the power rule to each logarithmic term.
2. Use the properties of logarithms to combine the terms into a single logarithm.

STEP 3

Apply the power rule to each term. The power rule states that aln⁡b=ln⁡ba a \ln b = \ln b^a .
For the first term: 7ln⁡x=ln⁡x7 7 \ln x = \ln x^7
For the second term: −18ln⁡y=ln⁡y−18 -\frac{1}{8} \ln y = \ln y^{-\frac{1}{8}}

STEP 4

Combine the terms using the properties of logarithms. Specifically, use the quotient rule which states that ln⁡a−ln⁡b=ln⁡(ab) \ln a - \ln b = \ln \left( \frac{a}{b} \right) .
Combine the terms: ln⁡x7−ln⁡y18=ln⁡(x7y18) \ln x^7 - \ln y^{\frac{1}{8}} = \ln \left( \frac{x^7}{y^{\frac{1}{8}}} \right)
The condensed expression is: ln⁡(x7y18) \boxed{\ln \left( \frac{x^7}{y^{\frac{1}{8}}} \right)}

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