Question

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

5e^(x-3) = 35

Answer 1

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

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

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

3. **Use the property of logarithms**: The property \( \ln(e^y) = y \) allows us to simplify the left side:
\[
x - 3 = \ln(7)
\]

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

5. **Final answer**: Thus, the solution to the equation is:
\[
x = \ln(7) + 3
\]

You can leave the answer in this form or calculate \( \ln(7) \) if you need a numerical approximation. The approximate value of \( \ln(7) \) is about 1.9459, so:
\[
x \approx 1.9459 + 3 \approx 4.9459
\]
Therefore, the final answer can be expressed as:
\[
x = \ln(7) + 3 \quad \text{or} \quad x \approx 4.9459
\]