To solve for [tex]\( y \)[/tex] in the equation [tex]\( 12^{9y} = 7 \)[/tex], we can use logarithms to simplify and solve the equation. Here's a detailed, step-by-step solution:
1. Take the natural logarithm (ln) of both sides of the equation, which will help us bring the exponent down:
[tex]\[
\ln(12^{9y}) = \ln(7)
\][/tex]
2. Use the power rule of logarithms to bring down the exponent [tex]\( 9y \)[/tex]:
[tex]\[
9y \cdot \ln(12) = \ln(7)
\][/tex]
3. Isolate [tex]\( y \)[/tex] by dividing both sides of the equation by [tex]\( 9 \cdot \ln(12) \)[/tex]:
[tex]\[
y = \frac{\ln(7)}{9 \cdot \ln(12)}
\][/tex]
4. Calculate the natural logarithms of the numbers involved.
[tex]\[
\ln(7) \approx 1.9459101490553132
\][/tex]
[tex]\[
\ln(12) \approx 2.4849066497880004
\][/tex]
5. Substitute these values back into the equation:
[tex]\[
y = \frac{1.9459101490553132}{9 \cdot 2.4849066497880004}
\][/tex]
6. Perform the multiplication in the denominator:
[tex]\[
9 \cdot 2.4849066497880004 = 22.364159848092
\][/tex]
7. Now, divide the numerator by the result of the denominator:
[tex]\[
y = \frac{1.9459101490553132}{22.364159848092} \approx 0.08701020571632734
\][/tex]
8. Round the result to the nearest hundredth:
[tex]\[
y \approx 0.09
\][/tex]
So, the value of [tex]\( y \)[/tex] rounded to the nearest hundredth is [tex]\( \boxed{0.09} \)[/tex].
