Question

solve the equation.
$e^{x - 1} = 9$
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. \\(x = \square\\)
(round to four decimal places as needed.)
\\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Take natural log of both sides

To solve \( e^{x - 1}=9 \), we take the natural logarithm (ln) of both sides. Recall that \( \ln(e^y)=y \) by the property of logarithms. So, applying \( \ln \) to both sides:
\( \ln(e^{x - 1})=\ln(9) \)
This simplifies to \( x - 1=\ln(9) \) because \( \ln(e^{x - 1}) = x - 1 \).

Step2: Solve for x

Now, we solve for \( x \) by adding 1 to both sides of the equation \( x - 1=\ln(9) \).
\( x=\ln(9)+1 \)
We know that \( \ln(9)\approx2.197224577 \) (using a calculator to find the natural logarithm of 9). Then, adding 1:
\( x\approx2.197224577 + 1=3.197224577 \)
Rounding to four decimal places, we get \( x\approx3.1972 \).

Answer:

A. \( x = 3.1972 \)