Question

solve for x to the nearest hundreth
9e^x=1

Answer 1

To solve the equation \( 9e^x = 1 \), we can follow these steps:

1. Divide both sides by 9:
\[
e^x = \frac{1}{9}
\]

2. Take the natural logarithm (ln) of both sides:
\[
\ln(e^x) = \ln\left(\frac{1}{9}\right)
\]

3. Use the property of logarithms \( \ln(e^x) = x \):
\[
x = \ln\left(\frac{1}{9}\right)
\]

4. Use the properties of logarithms to simplify:
\[
x = \ln(1) - \ln(9) = 0 - \ln(9) = -\ln(9)
\]

Now we can calculate \( -\ln(9) \) using a calculator:

\[
\ln(9) \approx 2.1972
\]
Thus,
\[
x \approx -2.1972
\]

Rounding to the nearest hundredth:
\[
x \approx -2.20
\]

So, the solution for \( x \) to the nearest hundredth is:
\[
\boxed{-2.20}
\]