To solve the equation [tex]\( 11^{9x} = 16^{x-8} \)[/tex] for [tex]\( x \)[/tex], we will use logarithms to isolate [tex]\( x \)[/tex].
1. Take the natural logarithm (or log base 10, but we'll use natural logarithm for this explanation) on both sides:
[tex]\[
\ln(11^{9x}) = \ln(16^{x-8})
\][/tex]
2. Use the property of logarithms that allows us to bring the exponent in front:
[tex]\[
9x \ln(11) = (x-8) \ln(16)
\][/tex]
3. Distribute the logarithm on the right-hand side:
[tex]\[
9x \ln(11) = x \ln(16) - 8 \ln(16)
\][/tex]
4. Move all terms involving [tex]\( x \)[/tex] to one side of the equation:
[tex]\[
9x \ln(11) - x \ln(16) = -8 \ln(16)
\][/tex]
5. Factor out [tex]\( x \)[/tex] on the left-hand side:
[tex]\[
x (9 \ln(11) - \ln(16)) = -8 \ln(16)
\][/tex]
6. Solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( (9 \ln(11) - \ln(16)) \)[/tex]:
[tex]\[
x = \frac{-8 \ln(16)}{9 \ln(11) - \ln(16)}
\][/tex]
Hence, the exact solution is:
[tex]\[
x = \frac{-8 \ln(16)}{9 \ln(11) - \ln(16)}
\][/tex]
This is the solution expressed in terms of natural logarithms.
