To solve the equation [tex]\( 7 + 3 \ln(t) = 15 \)[/tex], we need to isolate [tex]\( t \)[/tex] step-by-step. Here's how we can do it:
1. Isolate the logarithmic term:
[tex]\[
7 + 3 \ln(t) = 15
\][/tex]
Subtract 7 from both sides:
[tex]\[
3 \ln(t) = 15 - 7
\][/tex]
Simplify the right-hand side:
[tex]\[
3 \ln(t) = 8
\][/tex]
2. Solve for [tex]\( \ln(t) \)[/tex]:
Divide both sides by 3:
[tex]\[
\ln(t) = \frac{8}{3}
\][/tex]
3. Exponentiate both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
t = e^{\frac{8}{3}}
\][/tex]
Since [tex]\( \frac{8}{3} \approx 2.6666666666666665 \)[/tex], we calculate [tex]\( e^{\frac{8}{3}} \)[/tex].
4. Calculate the numerical value:
[tex]\[
t \approx 14.39
\][/tex]
So, the solution to the equation [tex]\( 7 + 3 \ln(t) = 15 \)[/tex] is approximately [tex]\( t = 14.39 \)[/tex]. The correct answer is:
[tex]\[
t = 14.39
\][/tex]
