Questions: Solve for (x). [3^-7 x=17^-x-5] Write the exact answer using either base-10 or base-e logarithms. [x=]

Solve for (x). [3^-7 x=17^-x-5] Write the exact answer using either base-10 or base-e logarithms. [x=]
Transcript text: Solve for $x$. \[ 3^{-7 x}=17^{-x-5} \] Write the exact answer using either base-10 or base-e logarithms. \[ x= \]
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Solution

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Solution Steps

Step 1: Take the Logarithm of Both Sides

We start with the equation: \[ 3^{-7x} = 17^{-x-5} \] Taking the logarithm of both sides, we can express the equation as: \[ -7x \log(3) = (-x - 5) \log(17) \]

Step 2: Rearrange the Equation

Rearranging the equation gives us: \[ -7x \log(3) = -x \log(17) - 5 \log(17) \] This can be rewritten as: \[ -7x \log(3) + x \log(17) = -5 \log(17) \]

Step 3: Factor Out \(x\)

Factoring out \(x\) from the left side results in: \[ x(\log(17) - 7 \log(3)) = -5 \log(17) \] Now, we can solve for \(x\): \[ x = \frac{-5 \log(17)}{\log(17) - 7 \log(3)} \]

Final Answer

Thus, the exact solution for \(x\) is: \[ \boxed{x = \frac{-5 \log(17)}{\log(17) - 7 \log(3)}} \]

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