Questions: Solve the exponential equation. Express the solution in terms of natural logarithms. Then, use a calculator to obtain a decimal approximation for the solution. e^x=20.4 What is the solution in terms of natural logarithms? The solution set is (Use a comma to separate answers as needed. Simplify your answer. Use integers or decimals for any numbers in the expression.)

Solution Steps

To solve the exponential equation \( e^x = 20.4 \), we need to take the natural logarithm of both sides. This will allow us to isolate \( x \). The natural logarithm of \( e^x \) is \( x \), so we get \( x = \ln(20.4) \). We can then use a calculator to find the decimal approximation of \( \ln(20.4) \).

1. Take the natural logarithm of both sides of the equation \( e^x = 20.4 \).
2. Simplify to isolate \( x \).
3. Use a calculator to find the decimal approximation of \( \ln(20.4) \).

To solve the equation \( e^x = 20.4 \), we take the natural logarithm of both sides:

\[ \ln(e^x) = \ln(20.4) \]

Using the property of logarithms that states \( \ln(e^x) = x \), we can simplify the equation to:

\[ x = \ln(20.4) \]

Calculating the natural logarithm of \( 20.4 \) gives us:

\[ x \approx 3.0155 \]

Final Answer