To solve for [tex]\( y \)[/tex] in the equation [tex]\( e^{y + 9} = 4 \)[/tex], we need to follow these steps:
1. Take the natural logarithm of both sides of the equation: The natural logarithm ([tex]\(\ln\)[/tex]) is the inverse function of the exponential function. This allows us to isolate the exponent [tex]\( y + 9 \)[/tex].
[tex]\[
\ln(e^{y + 9}) = \ln(4)
\][/tex]
2. Apply the property of logarithms: One important property of logarithms is that [tex]\(\ln(e^x) = x\)[/tex]. So, applying this property to the left-hand side of the equation we get:
[tex]\[
y + 9 = \ln(4)
\][/tex]
3. Isolate [tex]\( y \)[/tex]: We need to find the value of [tex]\( y \)[/tex], so we rearrange the equation to solve for [tex]\( y \)[/tex].
[tex]\[
y = \ln(4) - 9
\][/tex]
4. Calculate [tex]\(\ln(4)\)[/tex]: Using a calculator or mathematical table, we find:
[tex]\[
\ln(4) \approx 1.3862943611198906
\][/tex]
5. Substitute [tex]\(\ln(4)\)[/tex] into the equation: Now substitute this value back into our rearranged equation.
[tex]\[
y = 1.3862943611198906 - 9
\][/tex]
6. Perform the subtraction: So,
[tex]\[
y = 1.3862943611198906 - 9 = -7.613705638880109
\][/tex]
7. Round the result to the nearest hundredth: Finally, we round the computed value to the nearest hundredth.
[tex]\[
y \approx -7.61
\][/tex]
So, the value of [tex]\( y \)[/tex] rounded to the nearest hundredth is [tex]\( -7.61 \)[/tex].
