To solve for [tex]\( y \)[/tex] in the equation [tex]\( 8^{8y} = 15 \)[/tex], follow these steps:
1. Take the natural logarithm (ln) of both sides of the equation:
[tex]\[
\ln(8^{8y}) = \ln(15)
\][/tex]
2. Apply the power rule of logarithms, which states [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[
8y \cdot \ln(8) = \ln(15)
\][/tex]
3. Isolate the variable [tex]\( y \)[/tex] by dividing both sides of the equation by [tex]\( 8 \cdot \ln(8) \)[/tex]:
[tex]\[
y = \frac{\ln(15)}{8 \cdot \ln(8)}
\][/tex]
4. Calculate [tex]\(\ln(15)\)[/tex] and [tex]\(\ln(8)\)[/tex]:
[tex]\[
\ln(15) \approx 2.70805020110221
\][/tex]
[tex]\[
\ln(8) \approx 2.0794415416798357
\][/tex]
5. Plug these values into the equation:
[tex]\[
y = \frac{2.70805020110221}{8 \cdot 2.0794415416798357}
\][/tex]
6. Perform the multiplication in the denominator:
[tex]\[
8 \cdot 2.0794415416798357 \approx 16.635532333438686
\][/tex]
7. Now, divide the numerator by the newly computed denominator:
[tex]\[
y = \frac{2.70805020110221}{16.635532333438686} \approx 0.16278710815035496
\][/tex]
8. Finally, round the result to the nearest hundredth:
[tex]\[
y \approx 0.16
\][/tex]
Therefore, the value of [tex]\( y \)[/tex] is approximately [tex]\( 0.16 \)[/tex] when rounded to the nearest hundredth.
