To solve the equation [tex]\( 13^{-3x} = 12^{x+8} \)[/tex] for [tex]\( x \)[/tex], we should use logarithms. Here's a detailed, step-by-step solution:
1. Rewrite the equation using logarithms:
[tex]\[
13^{-3x} = 12^{x+8}
\][/tex]
2. Take the natural logarithm (or common logarithm) of both sides:
[tex]\[
\ln(13^{-3x}) = \ln(12^{x+8})
\][/tex]
3. Apply the power rule of logarithms [tex]\( \ln(a^b) = b \ln(a) \)[/tex]:
[tex]\[
-3x \ln(13) = (x + 8) \ln(12)
\][/tex]
4. Distribute the logarithm on the right-hand side:
[tex]\[
-3x \ln(13) = x \ln(12) + 8 \ln(12)
\][/tex]
5. Collect the terms involving [tex]\( x \)[/tex] on one side of the equation:
[tex]\[
-3x \ln(13) - x \ln(12) = 8 \ln(12)
\][/tex]
6. Factor out [tex]\( x \)[/tex] from the left side:
[tex]\[
x (-3 \ln(13) - \ln(12)) = 8 \ln(12)
\][/tex]
7. Solve for [tex]\( x \)[/tex] by isolating it on one side:
[tex]\[
x = \frac{8 \ln(12)}{-3 \ln(13) - \ln(12)}
\][/tex]
8. Simplify the expression if possible:
The exact solution for [tex]\( x \)[/tex] is:
[tex]\[
x = -\frac{8 \ln(12)}{\ln(26364)}
\][/tex]
So the final answer is:
[tex]\[
x = -8 \frac{\log(12)}{\log(26364)}
\][/tex]
