Answer :
Let's start by examining each expression one by one, focusing on expressing the logarithms in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex].
### 1. [tex]\(\ln \frac{81}{125}\)[/tex]
First, let's rewrite 81 and 125 as powers of their prime factors:
- [tex]\(81 = 3^4\)[/tex]
- [tex]\(125 = 5^3\)[/tex]
Thus, the expression becomes:
[tex]\[ \ln \frac{81}{125} = \ln \frac{3^4}{5^3} \][/tex]
Using the property of logarithms [tex]\(\ln \left( \frac{a}{b} \right) = \ln a - \ln b\)[/tex]:
[tex]\[ \ln \frac{3^4}{5^3} = \ln 3^4 - \ln 5^3 \][/tex]
Now, applying the power rule of logarithms, [tex]\(\ln a^b = b \ln a\)[/tex]:
[tex]\[ \ln 3^4 - \ln 5^3 = 4 \ln 3 - 3 \ln 5 \][/tex]
So,
[tex]\[ \ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5 \][/tex]
### 2. [tex]\(4 \ln 5 - 3 \ln 3\)[/tex]
This expression is already expressed in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
[tex]\[ 4 \ln 5 - 3 \ln 3 \][/tex]
No further manipulation is needed.
### 3. [tex]\(5 \ln 3 - 3 \ln 4\)[/tex]
To express this in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex], we need to rewrite [tex]\(\ln 4\)[/tex] in terms of these logarithms:
Since [tex]\(4 = 2^2\)[/tex],
[tex]\[ \ln 4 = \ln 2^2 = 2 \ln 2 \][/tex]
If more information about [tex]\(\ln 2\)[/tex] were given, we could substitute it into our expression. However, since we are asked only to use [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex], the expression remains in its current form for now:
[tex]\[ 5 \ln 3 - 3 \cdot 2 \ln 2 = 5 \ln 3 - 6 \ln 2 \][/tex]
### 4. [tex]\(4 \ln 3 - 3 \ln 5\)[/tex]
This expression is already expressed in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
[tex]\[ 4 \ln 3 - 3 \ln 5 \][/tex]
No further manipulation is needed.
### 5. [tex]\(3 \ln 4 - 5 \ln 3\)[/tex]
Again, expressing [tex]\(\ln 4\)[/tex] in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
Since [tex]\(4 = 2^2\)[/tex],
[tex]\[ 3 \ln 4 = 3 \ln 2^2 = 3 \cdot 2 \ln 2 = 6 \ln 2 \][/tex]
So, the expression is:
[tex]\[ 6 \ln 2 - 5 \ln 3 \][/tex]
If there were details connecting [tex]\(\ln 2\)[/tex] with [tex]\(\ln 3\)[/tex] or [tex]\(\ln 5\)[/tex], we could substitute those in. Since we lack that information, the expression remains unchanged.
### Summary
The expressions in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex] are:
1. [tex]\(\ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5\)[/tex]
2. [tex]\(4 \ln 5 - 3 \ln 3\)[/tex]
3. [tex]\(5 \ln 3 - 6 \ln 2\)[/tex]
4. [tex]\(4 \ln 3 - 3 \ln 5\)[/tex]
5. [tex]\(6 \ln 2 - 5 \ln 3\)[/tex]
Expressions 3 and 5 contain [tex]\(\ln 2\)[/tex], which cannot be simplified further without additional information about [tex]\(\ln 2\)[/tex]. The other expressions are already in the desired form.
### 1. [tex]\(\ln \frac{81}{125}\)[/tex]
First, let's rewrite 81 and 125 as powers of their prime factors:
- [tex]\(81 = 3^4\)[/tex]
- [tex]\(125 = 5^3\)[/tex]
Thus, the expression becomes:
[tex]\[ \ln \frac{81}{125} = \ln \frac{3^4}{5^3} \][/tex]
Using the property of logarithms [tex]\(\ln \left( \frac{a}{b} \right) = \ln a - \ln b\)[/tex]:
[tex]\[ \ln \frac{3^4}{5^3} = \ln 3^4 - \ln 5^3 \][/tex]
Now, applying the power rule of logarithms, [tex]\(\ln a^b = b \ln a\)[/tex]:
[tex]\[ \ln 3^4 - \ln 5^3 = 4 \ln 3 - 3 \ln 5 \][/tex]
So,
[tex]\[ \ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5 \][/tex]
### 2. [tex]\(4 \ln 5 - 3 \ln 3\)[/tex]
This expression is already expressed in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
[tex]\[ 4 \ln 5 - 3 \ln 3 \][/tex]
No further manipulation is needed.
### 3. [tex]\(5 \ln 3 - 3 \ln 4\)[/tex]
To express this in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex], we need to rewrite [tex]\(\ln 4\)[/tex] in terms of these logarithms:
Since [tex]\(4 = 2^2\)[/tex],
[tex]\[ \ln 4 = \ln 2^2 = 2 \ln 2 \][/tex]
If more information about [tex]\(\ln 2\)[/tex] were given, we could substitute it into our expression. However, since we are asked only to use [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex], the expression remains in its current form for now:
[tex]\[ 5 \ln 3 - 3 \cdot 2 \ln 2 = 5 \ln 3 - 6 \ln 2 \][/tex]
### 4. [tex]\(4 \ln 3 - 3 \ln 5\)[/tex]
This expression is already expressed in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
[tex]\[ 4 \ln 3 - 3 \ln 5 \][/tex]
No further manipulation is needed.
### 5. [tex]\(3 \ln 4 - 5 \ln 3\)[/tex]
Again, expressing [tex]\(\ln 4\)[/tex] in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex]:
Since [tex]\(4 = 2^2\)[/tex],
[tex]\[ 3 \ln 4 = 3 \ln 2^2 = 3 \cdot 2 \ln 2 = 6 \ln 2 \][/tex]
So, the expression is:
[tex]\[ 6 \ln 2 - 5 \ln 3 \][/tex]
If there were details connecting [tex]\(\ln 2\)[/tex] with [tex]\(\ln 3\)[/tex] or [tex]\(\ln 5\)[/tex], we could substitute those in. Since we lack that information, the expression remains unchanged.
### Summary
The expressions in terms of [tex]\(\ln 3\)[/tex] and [tex]\(\ln 5\)[/tex] are:
1. [tex]\(\ln \frac{81}{125} = 4 \ln 3 - 3 \ln 5\)[/tex]
2. [tex]\(4 \ln 5 - 3 \ln 3\)[/tex]
3. [tex]\(5 \ln 3 - 6 \ln 2\)[/tex]
4. [tex]\(4 \ln 3 - 3 \ln 5\)[/tex]
5. [tex]\(6 \ln 2 - 5 \ln 3\)[/tex]
Expressions 3 and 5 contain [tex]\(\ln 2\)[/tex], which cannot be simplified further without additional information about [tex]\(\ln 2\)[/tex]. The other expressions are already in the desired form.
