Question

solve the exponential equation. approximate your answer to the nearest thousandth. \\( e^{2.7x} = 6.12 \\) \\( x \approx \square \\) (type an integer or decimal rounded to the nearest thousandth as needed.)

Explanation:

Step1: Take natural log of both sides

To solve \( e^{2.7x} = 6.12 \), 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 gives:
\( \ln(e^{2.7x}) = \ln(6.12) \)

Step2: Simplify left side

Using the logarithm property \( \ln(e^y) = y \), the left side simplifies to \( 2.7x \). So we have:
\( 2.7x = \ln(6.12) \)

Step3: Solve for x

Divide both sides by 2.7 to isolate x:
\( x = \frac{\ln(6.12)}{2.7} \)

Now, calculate the value. First, find \( \ln(6.12) \approx 1.80017 \) (using a calculator). Then divide by 2.7:
\( x \approx \frac{1.80017}{2.7} \approx 0.66673 \)

Rounding to the nearest thousandth (three decimal places), we get \( x \approx 0.667 \).

Answer:

\( x \approx 0.667 \)