To solve for [tex]\(y\)[/tex] in the equation [tex]\(e^{-4y} = 7\)[/tex], follow these steps:
1. Take the natural logarithm of both sides to make the exponent more manageable.
[tex]\[
\ln(e^{-4y}) = \ln(7)
\][/tex]
2. Simplify the left-hand side using the property of logarithms [tex]\(\ln(e^x) = x\)[/tex].
[tex]\[
-4y = \ln(7)
\][/tex]
3. Solve for [tex]\(y\)[/tex] by isolating [tex]\(y\)[/tex]. To do this, divide both sides of the equation by [tex]\(-4\)[/tex].
[tex]\[
y = \frac{\ln(7)}{-4}
\][/tex]
4. Calculate [tex]\(\ln(7)\)[/tex] and then divide by [tex]\(-4\)[/tex] to find the value of [tex]\(y\)[/tex].
[tex]\[
\ln(7) \approx 1.945910
\][/tex]
Thus,
[tex]\[
y = \frac{1.945910}{-4} \approx -0.4864775372638283
\][/tex]
5. Round the answer to the nearest hundredth.
[tex]\[
y \approx -0.49
\][/tex]
So, the solution to the equation [tex]\(e^{-4y} = 7\)[/tex], rounded to the nearest hundredth, is [tex]\(y \approx -0.49\)[/tex].
