Question

solve the exponential equation. approximate your answer to the nearest thousandth.\\( e^{3.3x} = 7.56 \\)\\( x \approx \square \\)\\( \text{(type an integer or decimal rounded to the nearest thousandth as needed.)} \\)

Explanation:

Step1: Take natural log of both sides

To solve the exponential equation \( e^{3.3x} = 7.56 \), 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:
\( \ln(e^{3.3x}) = \ln(7.56) \)

Step2: Simplify left side

Using the logarithm property \( \ln(e^y) = y \), the left side simplifies to \( 3.3x \). So now we have:
\( 3.3x = \ln(7.56) \)

Step3: Solve for x

Divide both sides by \( 3.3 \) to isolate \( x \):
\( x = \frac{\ln(7.56)}{3.3} \)

Step4: Calculate the value

First, find \( \ln(7.56) \). Using a calculator, \( \ln(7.56) \approx 2.0296 \). Then divide by \( 3.3 \):
\( x \approx \frac{2.0296}{3.3} \approx 0.615 \)

Answer:

\( x \approx 0.615 \)