Sure, let's solve the given expression step-by-step:
Given:
[tex]\[
\log_x 2 - \left( \log 3 + \log_x x \right)
\][/tex]
First, let's break down each term in the expression.
### Expression Breakdown:
1. [tex]\(\log_x 2\)[/tex]: This is the logarithm of 2 with base [tex]\(x\)[/tex].
2. [tex]\(\log 3\)[/tex]: This is the logarithm of 3 with the default base, which is generally understood to be 10 (common log) or [tex]\(e\)[/tex] (natural log).
3. [tex]\(\log_x x\)[/tex]: This is the logarithm of [tex]\(x\)[/tex] with base [tex]\(x\)[/tex]. By the properties of logarithms, this is equal to:
[tex]\[\log_x x = 1\][/tex]
Now, let's plug these back into our expression:
[tex]\[
\log_x 2 - \left( \log 3 + 1 \right)
\][/tex]
### Simplifying Logarithms:
1. Logarithm base conversion:
Recall the change of base formula for logarithms:
[tex]\[
\log_b a = \frac{\log_k a}{\log_k b}
\][/tex]
This allows us to convert any logarithm to another base (commonly to base 10 or [tex]\(e\)[/tex]).
So, [tex]\(\log_x 2\)[/tex] can be written as:
[tex]\[\log_x 2 = \frac{\log(2)}{\log(x)}\][/tex]
Putting this into the expression:
[tex]\[
\frac{\log(2)}{\log(x)} - \left( \log 3 + 1 \right)
\][/tex]
### Writing out the full expression:
[tex]\[
\frac{\log(2)}{\log(x)} - \log(3) - 1
\][/tex]
### Final Expression:
After combining all terms, we get the simplified form:
[tex]\[
-\log(x) - \log(3) + \frac{\log(2)}{\log(x)}
\][/tex]
Therefore, the simplified form of the given expression:
[tex]\[
\log_x 2 - \left(\log 3 + \log_x x\right)
\][/tex]
is:
[tex]\[
-\log(x) - \log(3) + \frac{\log(2)}{\log(x)}
\][/tex]
