To express \(\ln \frac{81}{125}\) in terms of \(\ln 3\) and \(\ln 5\), follow these steps:
1. Apply the properties of logarithms: Use the property that \(\ln \frac{a}{b} = \ln a - \ln b\).
[tex]\[
\ln \frac{81}{125} = \ln 81 - \ln 125
\][/tex]
2. Prime factorize each number:
- \(81\) can be written as \(3^4\).
- \(125\) can be written as \(5^3\).
3. Substitute these factorizations into the logarithms:
[tex]\[
\ln 81 = \ln (3^4)
\][/tex]
[tex]\[
\ln 125 = \ln (5^3)
\][/tex]
4. Apply the power rule of logarithms: The power rule states that \(\ln (a^b) = b \ln a\).
[tex]\[
\ln (3^4) = 4 \ln 3
\][/tex]
[tex]\[
\ln (5^3) = 3 \ln 5
\][/tex]
5. Combine the results: Substitute these back into the original expression.
[tex]\[
\ln \frac{81}{125} = \ln 81 - \ln 125 = 4 \ln 3 - 3 \ln 5
\][/tex]
So, the logarithm \(\ln \frac{81}{125}\) expressed in terms of \(\ln 3\) and \(\ln 5\) is:
[tex]\[
\ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5
\][/tex]
