Let's solve each equation step-by-step and round the answers to the nearest hundredth.
### 1. Solving [tex]\( e^x = 2 \)[/tex]:
To solve for [tex]\( x \)[/tex] in the equation [tex]\( e^x = 2 \)[/tex], we can use logarithms. Specifically, we will use the natural logarithm (denoted as [tex]\( \ln \)[/tex]), which is the logarithm base [tex]\( e \)[/tex].
Step-by-step:
1. Take the natural logarithm of both sides:
[tex]\[ \ln(e^x) = \ln(2) \][/tex]
2. Apply the logarithm power rule (i.e., [tex]\( \ln(a^b) = b\ln(a) \)[/tex]) to simplify the left-hand side:
[tex]\[ x \cdot \ln(e) = \ln(2) \][/tex]
3. Since [tex]\( \ln(e) = 1 \)[/tex]:
[tex]\[ x = \ln(2) \][/tex]
Using precise mathematical tools, the value of [tex]\( \ln(2) \)[/tex] is approximately 0.693147. When we round this to the nearest hundredth, we get:
[tex]\[ x \approx 0.69 \][/tex]
### 2. Solving [tex]\( 5^{y+8} = 3 \)[/tex]:
To solve for [tex]\( y \)[/tex] in the equation [tex]\( 5^{y+8} = 3 \)[/tex], we use logarithms again, but this time we can use logarithms to any base. For simplicity, we'll use the logarithm to the base 5, denoted as [tex]\( \log_5 \)[/tex].
Step-by-step:
1. Take the logarithm base 5 of both sides:
[tex]\[ \log_5(5^{y+8}) = \log_5(3) \][/tex]
2. Apply the logarithm power rule (i.e., [tex]\( \log_b(a^c) = c\log_b(a) \)[/tex]) to simplify the left-hand side:
[tex]\[ (y+8) \cdot \log_5(5) = \log_5(3) \][/tex]
3. Since [tex]\( \log_5(5) = 1 \)[/tex]:
[tex]\[ y + 8 = \log_5(3) \][/tex]
4. To isolate [tex]\( y \)[/tex], subtract 8 from both sides:
[tex]\[ y = \log_5(3) - 8 \][/tex]
The value of [tex]\( \log_5(3) \)[/tex] is approximately 0.682606. Subsequently:
[tex]\[ \log_5(3) - 8 \approx 0.682606 - 8 = -7.317394 \][/tex]
When we round this to the nearest hundredth, we get:
[tex]\[ y \approx -7.32 \][/tex]
### Final Answers:
[tex]\[
\begin{array}{l}
x \approx 0.69 \\
y \approx -7.32
\end{array}
\][/tex]
