How do you solve #9^(8x)=13^(-x+7)#?

You need to use logarithms:

It is a very ugly solution.

To solve the equation (9^{8x} = 13^{-x+7}), we can take the natural logarithm (ln) of both sides to remove the exponents. This allows us to solve for the variable (x).

First, take the natural logarithm of both sides:

[ \ln(9^{8x}) = \ln(13^{-x+7}) ]

Using the properties of logarithms, we can bring down the exponents:

[ 8x \ln(9) = (-x+7) \ln(13) ]

Next, distribute the natural logarithms:

[ 8x \ln(9) = -x \ln(13) + 7 \ln(13) ]

Now, isolate the variable (x) by bringing all terms containing (x) to one side of the equation:

[ 8x \ln(9) + x \ln(13) = 7 \ln(13) ]

Factor out the common factor of (x):

[ x(8 \ln(9) + \ln(13)) = 7 \ln(13) ]

Finally, solve for (x) by dividing both sides by the coefficient of (x):

[ x = \frac{7 \ln(13)}{8 \ln(9) + \ln(13)} ]

This is the solution for the equation (9^{8x} = 13^{-x+7}).