To solve the equation \(2^{3x-6} = 9^{5x-5}\), we can take the logarithm of both sides to bring down the exponents. This will allow us to solve for \(x\) by isolating it on one side of the equation. We can use the properties of logarithms to simplify the expression and solve for \(x\).
The given equation is:
\[
2^{3x-6} = 9^{5x-5}
\]
To solve for \(x\), we can take the logarithm of both sides. Let's use the natural logarithm (\(\ln\)):
\[
\ln(2^{3x-6}) = \ln(9^{5x-5})
\]
Using the power rule of logarithms, \(\ln(a^b) = b \ln(a)\), we can rewrite both sides:
\[
(3x-6) \ln(2) = (5x-5) \ln(9)
\]
Expand both sides:
\[
3x \ln(2) - 6 \ln(2) = 5x \ln(9) - 5 \ln(9)
\]
Rearrange the terms to isolate the terms involving \(x\) on one side:
\[
3x \ln(2) - 5x \ln(9) = 6 \ln(2) - 5 \ln(9)
\]
Factor out \(x\) from the left side:
\[
x (3 \ln(2) - 5 \ln(9)) = 6 \ln(2) - 5 \ln(9)
\]
Divide both sides by \((3 \ln(2) - 5 \ln(9))\) to solve for \(x\):
\[
x = \frac{6 \ln(2) - 5 \ln(9)}{3 \ln(2) - 5 \ln(9)}
\]
Now, calculate the numerical value using a calculator:
- \(\ln(2) \approx 0.6931\)
- \(\ln(9) = \ln(3^2) = 2 \ln(3) \approx 2 \times 1.0986 = 2.1972\)
Substitute these values into the equation:
\[
x = \frac{6 \times 0.6931 - 5 \times 2.1972}{3 \times 0.6931 - 5 \times 2.1972}
\]
Calculate the numerator and the denominator:
- Numerator: \(6 \times 0.6931 - 5 \times 2.1972 = 4.1586 - 10.986 = -6.8274\)
- Denominator: \(3 \times 0.6931 - 5 \times 2.1972 = 2.0793 - 10.986 = -8.9067\)
Finally, calculate \(x\):
\[
x = \frac{-6.8274}{-8.9067} \approx 0.7665
\]
The solution to the equation is:
\[
\boxed{x = 0.7665}
\]
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