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

[tex]\[ e^{-4y} = 7 \][/tex]



Answer :

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].