Question

Solve the following exponential equation using logarithms. Remember to check your solution(s).
e^(5m+3) +9 =13

Answer 1

To solve the equation \( e^{5m+3} + 9 = 13 \), we first isolate the exponential term:

\[
e^{5m+3} = 13 - 9
\]

\[
e^{5m+3} = 4
\]

Next, we take the natural logarithm of both sides:

\[
\ln(e^{5m+3}) = \ln(4)
\]

Using the property of logarithms that \( \ln(e^x) = x \):

\[
5m + 3 = \ln(4)
\]

Now, we solve for \( m \):

\[
5m = \ln(4) - 3
\]

\[
m = \frac{\ln(4) - 3}{5}
\]

Now we can compute the value of \( m \):

\[
m = \frac{\ln(4) - 3}{5}
\]

Using a calculator to approximate \( \ln(4) \):

\[
\ln(4) \approx 1.386
\]

Substituting this value in:

\[
m \approx \frac{1.386 - 3}{5}
\]

\[
m \approx \frac{-1.614}{5} \approx -0.3228
\]

Now we will check our solution by substituting \( m \) back into the original equation:

1. Calculate \( 5m + 3 \):
\[
5(-0.3228) + 3 \approx -1.614 + 3 = 1.386
\]

2. Calculate \( e^{5m+3} \):
\[
e^{1.386} \approx 4
\]

3. Verify the original equation:
\[
e^{1.386} + 9 \approx 4 + 9 = 13
\]

Since both sides are equal, our solution is confirmed.

Thus, the final answer for \( m \) is:

\[
m = \frac{\ln(4) - 3}{5} \approx -0.3228
\]