Answer :
Let's solve the given equations step-by-step.
### Equation 1: [tex]\(12^{x+7} = 3\)[/tex]
1. Take the natural logarithm (ln) of both sides to simplify the equation:
[tex]\[ \ln(12^{x+7}) = \ln(3) \][/tex]
2. Use the logarithmic property [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[ (x+7) \ln(12) = \ln(3) \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 7 = \frac{\ln(3)}{\ln(12)} \][/tex]
4. Isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\ln(3)}{\ln(12)} - 7 \][/tex]
5. Calculate the natural logarithms:
[tex]\[ \ln(3) \approx 1.0986, \quad \ln(12) \approx 2.4849 \][/tex]
6. Plug in the values:
[tex]\[ x = \frac{1.0986}{2.4849} - 7 \][/tex]
7. Perform the division:
[tex]\[ \frac{1.0986}{2.4849} \approx 0.442 \][/tex]
8. Subtract 7:
[tex]\[ x = 0.442 - 7 \approx -6.558 \][/tex]
Rounded to the nearest hundredth:
[tex]\[ x \approx -6.56 \][/tex]
### Equation 2: [tex]\(e^y = 9\)[/tex]
1. Take the natural logarithm (ln) of both sides:
[tex]\[ \ln(e^y) = \ln(9) \][/tex]
2. Use the logarithmic property [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[ y \ln(e) = \ln(9) \][/tex]
3. Knowing that [tex]\(\ln(e) = 1\)[/tex]:
[tex]\[ y = \ln(9) \][/tex]
4. Calculate the natural logarithm:
[tex]\[ \ln(9) \approx 2.1972 \][/tex]
Rounded to the nearest hundredth:
[tex]\[ y \approx 2.20 \][/tex]
### Final Answers
[tex]\[ \begin{array}{l} x \approx -6.56 \\ y \approx 2.20 \end{array} \][/tex]
### Equation 1: [tex]\(12^{x+7} = 3\)[/tex]
1. Take the natural logarithm (ln) of both sides to simplify the equation:
[tex]\[ \ln(12^{x+7}) = \ln(3) \][/tex]
2. Use the logarithmic property [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[ (x+7) \ln(12) = \ln(3) \][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + 7 = \frac{\ln(3)}{\ln(12)} \][/tex]
4. Isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\ln(3)}{\ln(12)} - 7 \][/tex]
5. Calculate the natural logarithms:
[tex]\[ \ln(3) \approx 1.0986, \quad \ln(12) \approx 2.4849 \][/tex]
6. Plug in the values:
[tex]\[ x = \frac{1.0986}{2.4849} - 7 \][/tex]
7. Perform the division:
[tex]\[ \frac{1.0986}{2.4849} \approx 0.442 \][/tex]
8. Subtract 7:
[tex]\[ x = 0.442 - 7 \approx -6.558 \][/tex]
Rounded to the nearest hundredth:
[tex]\[ x \approx -6.56 \][/tex]
### Equation 2: [tex]\(e^y = 9\)[/tex]
1. Take the natural logarithm (ln) of both sides:
[tex]\[ \ln(e^y) = \ln(9) \][/tex]
2. Use the logarithmic property [tex]\(\ln(a^b) = b \ln(a)\)[/tex]:
[tex]\[ y \ln(e) = \ln(9) \][/tex]
3. Knowing that [tex]\(\ln(e) = 1\)[/tex]:
[tex]\[ y = \ln(9) \][/tex]
4. Calculate the natural logarithm:
[tex]\[ \ln(9) \approx 2.1972 \][/tex]
Rounded to the nearest hundredth:
[tex]\[ y \approx 2.20 \][/tex]
### Final Answers
[tex]\[ \begin{array}{l} x \approx -6.56 \\ y \approx 2.20 \end{array} \][/tex]
