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

You need to use logarithms:

It is a very ugly solution.

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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 $9^{8x} = 13^{-x+7}$.

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