To solve for [tex]\( y \)[/tex] in the equation [tex]\( 15^{9y} = 2 \)[/tex], follow these steps:
1. Take the natural logarithm (ln) of both sides:
[tex]\[
\ln(15^{9y}) = \ln(2)
\][/tex]
2. Use the property of logarithms that states [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[
9y \cdot \ln(15) = \ln(2)
\][/tex]
3. Solve for [tex]\( y \)[/tex] by isolating it on one side of the equation:
[tex]\[
y = \frac{\ln(2)}{9 \cdot \ln(15)}
\][/tex]
4. Substitute the known values for [tex]\(\ln(2)\)[/tex] and [tex]\(\ln(15)\)[/tex]:
[tex]\[
\ln(2) \approx 0.6931471805599453
\][/tex]
[tex]\[
\ln(15) \approx 2.70805020110221
\][/tex]
5. Calculate the denominator:
[tex]\[
9 \cdot \ln(15) \approx 9 \cdot 2.70805020110221 = 24.37245180991989
\][/tex]
6. Calculate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{0.6931471805599453}{24.37245180991989} \approx 0.02843978053442394
\][/tex]
7. Round the result for [tex]\( y \)[/tex] to the nearest hundredth:
[tex]\[
y \approx 0.03
\][/tex]
Thus, the value of [tex]\( y \)[/tex] rounded to the nearest hundredth is [tex]\(\boxed{0.03}\)[/tex].
