Sure! Let's take the expression [tex]\(\ln \frac{81}{125}\)[/tex] and express it in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
1. Rewrite as a difference of logarithms:
[tex]\[
\ln \frac{81}{125} = \ln 81 - \ln 125
\][/tex]
2. Express the arguments as powers:
Notice that [tex]\(81 = 3^4\)[/tex] and [tex]\(125 = 5^3\)[/tex]:
[tex]\[
\ln 81 = \ln (3^4) \quad \text{and} \quad \ln 125 = \ln (5^3)
\][/tex]
3. Apply the power rule of logarithms:
[tex]\(\ln (a^b) = b \ln a\)[/tex], so:
[tex]\[
\ln (3^4) = 4 \ln 3 \quad \text{and} \quad \ln (5^3) = 3 \ln 5
\][/tex]
4. Substitute these back into the original expression:
[tex]\[
\ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5
\][/tex]
Thus, [tex]\(\ln \frac{81}{125}\)[/tex] can be expressed in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex] as:
[tex]\[
\ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5
\][/tex]
Given the expressions provided, the correct one is:
[tex]\[
4 \ln 3 - 3 \ln 5
\][/tex]
So, the correct expression in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex] is:
[tex]\[
4 \ln 3 - 3 \ln 5
\][/tex]
