Question 6

Solve for $x$ :
\[
\begin{array}{l}
8^{2 x-10}=9^{8 x-2} \\
x=\square
\end{array}
\]

You may enter the exact value or round to 4 decimal places.
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Answer

$\boxed{x = \log\left(\frac{32768}{9}\right)^{\frac{1}{\log\left(\frac{8}{6561}\right)}}}$

Steps

Step 1 ：Given the equation $8^{2x - 10} = 9^{8x - 2}$

Step 2 ：Take the natural logarithm on both sides to get $(2x - 10) \cdot \ln(8) = (8x - 2) \cdot \ln(9)$

Step 3 ：Solve for x to get $x = \log\left(\frac{32768}{9}\right)^{\frac{1}{\log\left(\frac{8}{6561}\right)}}$

Step 4 ：$\boxed{x = \log\left(\frac{32768}{9}\right)^{\frac{1}{\log\left(\frac{8}{6561}\right)}}}$