Certainly! To express [tex]\(\ln 6\)[/tex] in terms of [tex]\(s\)[/tex] and [tex]\(t\)[/tex], we can use the properties of logarithms. Here is a detailed, step-by-step solution:
### Step-by-Step Solution
1. Given:
- [tex]\(\ln 2 = s\)[/tex]
- [tex]\(\ln 3 = t\)[/tex]
2. Expression to Find:
- [tex]\(\ln 6\)[/tex]
3. Using Properties of Logarithms:
- One of the key properties of logarithms is that the logarithm of a product is equal to the sum of the logarithms of the factors. This can be written as:
[tex]\[
\ln(a \cdot b) = \ln a + \ln b
\][/tex]
4. Applying this Property:
- Notice that [tex]\(6\)[/tex] can be expressed as the product of [tex]\(2\)[/tex] and [tex]\(3\)[/tex]:
[tex]\[
6 = 2 \times 3
\][/tex]
5. Rewrite [tex]\(\ln 6\)[/tex]:
- Using the property of logarithms mentioned, we can rewrite [tex]\(\ln 6\)[/tex] as:
[tex]\[
\ln 6 = \ln (2 \times 3) = \ln 2 + \ln 3
\][/tex]
6. Substituting the Given Values:
- We know from the given information that [tex]\(\ln 2 = s\)[/tex] and [tex]\(\ln 3 = t\)[/tex]. So, we can substitute these values into the equation:
[tex]\[
\ln 6 = \ln 2 + \ln 3 = s + t
\][/tex]
Therefore, [tex]\(\ln 6\)[/tex] in terms of [tex]\(s\)[/tex] and [tex]\(t\)[/tex] is:
[tex]\[
\ln 6 = s + t
\][/tex]
When [tex]\(s = \ln 2\)[/tex] and [tex]\(t = \ln 3\)[/tex], the logarithm of [tex]\(6\)[/tex] can be expressed as the sum of [tex]\(s\)[/tex] and [tex]\(t\)[/tex]:
[tex]\[
\ln 6 = s + t = 1.791759469228055
\][/tex]
This completes our solution, explaining how [tex]\(\ln 6\)[/tex] is related to [tex]\(s\)[/tex] and [tex]\(t\)[/tex].
