Alright, let's proceed by expressing each logarithm in terms of \(\ln 3\) and \(\ln 5\).
### 1. \(\ln \frac{81}{125}\)
The term \(\frac{81}{125}\) can be broken down into:
[tex]\[ \ln \frac{81}{125} = \ln 81 - \ln 125 \][/tex]
Next, let's express 81 and 125 in terms of their prime factors:
[tex]\[ 81 = 3^4 \][/tex]
[tex]\[ 125 = 5^3 \][/tex]
So, we have:
[tex]\[ \ln 81 = \ln (3^4) = 4 \ln 3 \][/tex]
[tex]\[ \ln 125 = \ln (5^3) = 3 \ln 5 \][/tex]
Combining these, we get:
[tex]\[ \ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5 \][/tex]
### 2. \(4 \ln 5 - 3 \ln 3\)
This expression is already in terms of \(\ln 3\) and \(\ln 5\). Nothing else needs to be done:
[tex]\[ 4 \ln 5 - 3 \ln 3 \][/tex]
### 3. \(5 \ln 3 - 3 \ln 4\)
We observe that \( \ln 4 \) can be expressed using logarithms of 2:
[tex]\[ \ln 4 = \ln (2^2) = 2 \ln 2 \][/tex]
Unfortunately, we do not have \(\ln 2\) in our given basis (\(\ln 3\) and \(\ln 5\)), so this expression is as simplified as it can get in terms of the given bases:
[tex]\[ 5 \ln 3 - 3 \ln 4 \][/tex]
### 4. \(4 \ln 3 - 3 \ln 5\)
This expression is already in terms of \(\ln 3\) and \(\ln 5\). No further modifications are needed:
[tex]\[ 4 \ln 3 - 3 \ln 5 \][/tex]
### 5. \(3 \ln 4 - 5 \ln 3\)
Again, we can express \( \ln 4 \) in terms of \(\ln 2\):
[tex]\[ \ln 4 = 2 \ln 2 \][/tex]
Given we do not have \(\ln 2\) in our basis (\(\ln 3\) and \(\ln 5\)), we leave this as:
[tex]\[ 3 \ln 4 - 5 \ln 3 \][/tex]
Now let's summarize the results and provide their numerical values:
### Numerical Results:
1. \(\ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5 \approx -0.4339\)
2. \(4 \ln 5 - 3 \ln 3 \approx 3.1419\)
3. \(5 \ln 3 - 3 \ln 4 \approx 1.3342\)
4. \(4 \ln 3 - 3 \ln 5 \approx -0.4339\)
5. \(3 \ln 4 - 5 \ln 3 \approx -1.3342\)
Thus, we have expressed all given logarithms in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex], and provided their approximate numerical values.
