To solve the equation [tex]\( 13^{-3x} = 12^{x+8} \)[/tex], we will use logarithms to isolate [tex]\( x \)[/tex]. Here is the detailed step-by-step solution:
1. Express both sides with logarithms:
We will take the natural logarithm (or logarithm base 10) of both sides to use the properties of logarithms:
[tex]\[
\ln(13^{-3x}) = \ln(12^{x+8})
\][/tex]
2. Apply the power rule of logarithms:
The power rule [tex]\(\ln(a^b) = b \ln(a)\)[/tex] allows us to bring the exponents down as multipliers:
[tex]\[
-3x \ln(13) = (x + 8) \ln(12)
\][/tex]
3. Distribute [tex]\(\ln(12)\)[/tex]:
[tex]\[
-3x \ln(13) = x \ln(12) + 8 \ln(12)
\][/tex]
4. Isolate the terms involving [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], let's get all the terms involving [tex]\( x \)[/tex] on one side and the constant terms on the other side:
[tex]\[
-3x \ln(13) - x \ln(12) = 8 \ln(12)
\][/tex]
5. Factor out [tex]\( x \)[/tex] from the left-hand side:
[tex]\[
x(-3 \ln(13) - \ln(12)) = 8 \ln(12)
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Divide both sides by the coefficient of [tex]\( x \)[/tex]:
[tex]\[
x = \frac{8 \ln(12)}{-3 \ln(13) - \ln(12)}
\][/tex]
Thus, the exact solution for [tex]\( x \)[/tex] in terms of natural logarithms (base [tex]\( e \)[/tex]) is:
[tex]\[
x = \frac{8 \ln(12)}{-3 \ln(13) - \ln(12)}
\][/tex]
