The problem asks to solve the equation $83 = 100 - 100e^{-0.07x}$ for $x$, and provide the answer correct to 3 decimal places.

1. Problem Description

The problem asks to solve the equation $83 = 100 - 100e^{-0.07x}$ for $x$, and provide the answer correct to 3 decimal places.

2. Solution Steps

We are given the equation $83 = 100 - 100e^{-0.07x}$.

First, isolate the exponential term:

$83 - 100 = -100e^{-0.07x}$

$-17 = -100e^{-0.07x}$

Divide both sides by $-100$:

$\frac{-17}{-100} = e^{-0.07x}$

$0.17 = e^{-0.07x}$

Now, take the natural logarithm of both sides:

$\ln(0.17) = \ln(e^{-0.07x})$

$\ln(0.17) = -0.07x$

Now, solve for $x$:

$x = \frac{\ln(0.17)}{-0.07}$

Using a calculator:

$\ln(0.17) \approx -1.7719566$

$x \approx \frac{-1.7719566}{-0.07}$

$x \approx 25.3136657$

Rounding to 3 decimal places:

$x \approx 25.314$

3. Final Answer

$x = 25.314$