To solve the inequality [tex]\(\frac{1}{\log_3 \pi} + \frac{1}{\log_4 \pi} > x\)[/tex], let's proceed step-by-step.
1. Understand the Inequality:
The expression [tex]\(\frac{1}{\log_3 \pi}\)[/tex] represents the reciprocal of the logarithm of [tex]\(\pi\)[/tex] with base 3, and similarly, [tex]\(\frac{1}{\log_4 \pi}\)[/tex] represents the reciprocal of the logarithm of [tex]\(\pi\)[/tex] with base 4.
2. Change of Base Formula:
We can use the logarithmic change of base formula to convert these expressions into a simpler form. Recall that for any logarithms, [tex]\(\log_a b = \frac{\log_c b}{\log_c a}\)[/tex].
3. Simplifying [tex]\(\frac{1}{\log_3 \pi}\)[/tex]:
Using the change of base formula:
[tex]\[
\frac{1}{\log_3 \pi} = \frac{1}{\frac{\log \pi}{\log 3}} = \frac{\log 3}{\log \pi}
\][/tex]
4. Simplifying [tex]\(\frac{1}{\log_4 \pi}\)[/tex]:
Similarly,
[tex]\[
\frac{1}{\log_4 \pi} = \frac{1}{\frac{\log \pi}{\log 4}} = \frac{\log 4}{\log \pi}
\][/tex]
5. Combining the Results:
Adding these two components together:
[tex]\[
\frac{\log 3}{\log \pi} + \frac{\log 4}{\log \pi}
\][/tex]
6. Factoring Out the Common Denominator:
Factor out the common denominator [tex]\(\log \pi\)[/tex]:
[tex]\[
\frac{\log 3 + \log 4}{\log \pi}
\][/tex]
7. Using Logarithm Properties:
Recall the logarithm property [tex]\(\log a + \log b = \log(ab)\)[/tex]:
[tex]\[
\frac{\log (3 \cdot 4)}{\log \pi} = \frac{\log 12}{\log \pi}
\][/tex]
8. Interpreting the Inequality:
The inequality now reads:
[tex]\[
\frac{\log 12}{\log \pi} > x
\][/tex]
9. Conclusion:
Therefore, [tex]\(x\)[/tex] must be less than the value of [tex]\(\frac{\log 12}{\log \pi}\)[/tex]. Given the precise numerical evaluation, we deduce:
[tex]\[
\frac{\log 12}{\log \pi} \approx 1.8677261107283452
\][/tex]
Hence, [tex]\(x\)[/tex] can be any value less than approximately 1.8677261107283452.
