Questions: Solve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal places. Use e = 2.71828182845905 e^(2x) = 80^(1+x)

Solution Steps

To solve the exponential equation \( e^{2x} = 80^{1+x} \), we can take the natural logarithm of both sides to simplify the equation. This will allow us to solve for \( x \) by isolating it on one side of the equation. After finding the exact expression for \( x \), we can compute its decimal approximation.

We start with the equation:

\[ e^{2x} = 80^{1+x} \]

Taking the natural logarithm of both sides gives us:

\[ \ln(e^{2x}) = \ln(80^{1+x}) \]

Using the properties of logarithms, we can simplify both sides:

\[ 2x = (1+x) \ln(80) \]

Rearranging the equation to isolate \( x \):

\[ 2x = \ln(80) + x \ln(80) \]

This can be rewritten as:

\[ 2x - x \ln(80) = \ln(80) \]

Factoring out \( x \):

\[ x(2 - \ln(80)) = \ln(80) \]

Thus, we find:

\[ x = \frac{\ln(80)}{2 - \ln(80)} \]

From our calculations, we have:

\[ \ln(80) \approx 4.3820 \]

Substituting \( \ln(80) \) into the equation for \( x \):

\[ x \approx \frac{4.3820}{2 - 4.3820} = \frac{4.3820}{-2.3820} \approx -1.8396 \]

Rounding \( x \) to two decimal places gives:

\[ x \approx -1.84 \]

Final Answer