We are asked to solve the equation $83 = 100 - 100e^{-0.9x}$ for $x$, and round the answer to three decimal places.

1. Problem Description

We are asked to solve the equation $83 = 100 - 100e^{-0.9x}$ for $x$, and round the answer to three decimal places.

2. Solution Steps

First, isolate the exponential term:

$83 = 100 - 100e^{-0.9x}$

$83 - 100 = -100e^{-0.9x}$

$-17 = -100e^{-0.9x}$

$\frac{-17}{-100} = e^{-0.9x}$

$0.17 = e^{-0.9x}$

Next, take the natural logarithm of both sides:

$\ln(0.17) = \ln(e^{-0.9x})$

$\ln(0.17) = -0.9x$

Finally, solve for $x$:

$x = \frac{\ln(0.17)}{-0.9}$

$x = \frac{-1.7719566}{-0.9}$

$x \approx 1.9688406$

Rounding to three decimal places, we get $x \approx 1.969$.

3. Final Answer

$x = 1.969$