To solve the equation [tex]\(7^{-x+5} = 13^{-6x}\)[/tex] for [tex]\(x\)[/tex], we will utilize the properties of logarithms. Here's the step-by-step process:
1. Take the Natural Logarithm of Both Sides:
[tex]\[
\ln(7^{-x+5}) = \ln(13^{-6x})
\][/tex]
2. Apply the Power Rule of Logarithms:
The power rule states [tex]\(\ln(a^b) = b \ln(a)\)[/tex]. Therefore, we can rewrite our equation as:
[tex]\[
(-x+5) \ln(7) = (-6x) \ln(13)
\][/tex]
3. Expand and Simplify the Equation:
Distribute the logarithms on both sides:
[tex]\[
-x \ln(7) + 5 \ln(7) = -6x \ln(13)
\][/tex]
4. Rearrange to Isolate Terms Involving [tex]\(x\)[/tex]:
Bring all the terms involving [tex]\(x\)[/tex] on one side:
[tex]\[
5 \ln(7) = x \ln(7) - 6x \ln(13)
\][/tex]
5. Factor Out [tex]\(x\)[/tex] from the Right Side:
[tex]\[
5 \ln(7) = x (\ln(7) - 6 \ln(13))
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\((\ln(7) - 6 \ln(13))\)[/tex]:
[tex]\[
x = \frac{5 \ln(7)}{\ln(7) - 6 \ln(13)}
\][/tex]
Therefore, the exact value of [tex]\(x\)[/tex] in terms of natural logarithms (base-e) is:
[tex]\[
x = \frac{5 \ln(7)}{\ln(7) - 6 \ln(13)}
\][/tex]
After evaluating this expression numerically, [tex]\(x\)[/tex] is approximately:
[tex]\[
x \approx -0.7237210372419273
\][/tex]
To summarize, the exact solution for [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{5 \ln(7)}{\ln(7) - 6 \ln(13)}
\][/tex]
