Solve for [tex]$y$[/tex] in the equation below. Round your answer to the nearest hundredth. Do not round any intermediate computations.

[tex]\[
8^{8y} = 15
\][/tex]



Answer :

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.