To solve the equation [tex]\(6^{-9x} = 7^{-x + 4}\)[/tex] for [tex]\(x\)[/tex], we need to use logarithms to linearize the exponents. Here are the steps involved:
1. Starting with the original equation:
[tex]\[
6^{-9x} = 7^{-x + 4}
\][/tex]
2. Apply the natural logarithm (or common logarithm) to both sides of the equation:
[tex]\[
\ln(6^{-9x}) = \ln(7^{-x + 4})
\][/tex]
3. Use the power rule of logarithms, [tex]\(\ln(a^b) = b \ln(a)\)[/tex], to bring the exponents in front:
[tex]\[
-9x \ln(6) = (-x + 4) \ln(7)
\][/tex]
4. Distribute the logarithm on the right-hand side:
[tex]\[
-9x \ln(6) = -x \ln(7) + 4 \ln(7)
\][/tex]
5. Isolate the term involving [tex]\(x\)[/tex] on one side of the equation:
[tex]\[
-9x \ln(6) + x \ln(7) = 4 \ln(7)
\][/tex]
6. Factor out [tex]\(x\)[/tex] from the left-hand side:
[tex]\[
x(-9 \ln(6) + \ln(7)) = 4 \ln(7)
\][/tex]
7. Solve for [tex]\(x\)[/tex] by dividing both sides by the coefficient of [tex]\(x\)[/tex]:
[tex]\[
x = \frac{4 \ln(7)}{-9 \ln(6) + \ln(7)}
\][/tex]
This is the exact solution for [tex]\(x\)[/tex] using logarithms. If required, the logarithms can be evaluated to their decimal approximations using a calculator, but the above expression is the precise answer.
