Solve the equation using any method. Round to the nearest thousandth.

Question

5e(x−3)=35
5
𝑒
(
𝑥
−
3
)
=
35

(1 point)
Responses

2.639
2.639

5.123
5.123

4.946
4.946

3.710

Answer 1

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

1. Isolate the exponential term: \[ e^{(x-3)} = \frac{35}{5} \] \[ e^{(x-3)} = 7 \]

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

3. Simplify using the property of logarithms \( \ln(e^y) = y \): \[ x - 3 = \ln(7) \]

4. Solve for \( x \): \[ x = \ln(7) + 3 \]

5. Calculate \( \ln(7) \): Using a calculator, we find: \[ \ln(7) \approx 1.945910 \]

6. Now substitute back into the equation for \( x \): \[ x \approx 1.945910 + 3 \approx 4.945910 \]

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

Thus, the solution to the equation \( 5e^{(x-3)} = 35 \) is \( \boxed{4.946} \).