Question

Solve the equation as simply as you can and write the steps.

5e^(x-3) = 35

Round to the nearest thounsandth

Answer 1

To solve the equation \( 5e^{x-3} = 35 \), follow these steps:

1. **Isolate \( e^{x-3} \)**:
Divide both sides of the equation by 5.
\[
e^{x-3} = \frac{35}{5}
\]
\[
e^{x-3} = 7
\]

2. **Take the natural logarithm of both sides**:
To eliminate the exponential, take the natural logarithm (ln) of both sides.
\[
\ln(e^{x-3}) = \ln(7)
\]

3. **Use the property of logarithms**:
The logarithm of an exponential function simplifies:
\[
x-3 = \ln(7)
\]

4. **Solve for \( x \)**:
Add 3 to both sides of the equation.
\[
x = \ln(7) + 3
\]

5. **Calculate \( \ln(7) \)**:
Use a calculator to compute \( \ln(7) \).
\[
\ln(7) \approx 1.945910
\]

Then substitute back to find \( x \):
\[
x \approx 1.945910 + 3
\]
\[
x \approx 4.945910
\]

6. **Round to the nearest thousandth**:
\[
x \approx 4.946
\]

Thus, the solution is:
\[
\boxed{4.946}
\]