Questions: Solve the given exponential equation by hand. Your answer will be e^t = 2 t = .6931

Solution Steps

To solve the exponential equation \( e^t = 2 \), we need to find the value of \( t \) that satisfies this equation. The natural logarithm function, denoted as \(\ln\), is the inverse of the exponential function with base \( e \). Therefore, we can take the natural logarithm of both sides of the equation to solve for \( t \).

1. Take the natural logarithm of both sides of the equation \( e^t = 2 \).
2. Use the property of logarithms that \(\ln(e^t) = t\).
3. Solve for \( t \) by calculating \(\ln(2)\).

We start with the equation: \[ e^t = 2 \] To solve for \( t \), we take the natural logarithm of both sides: \[ \ln(e^t) = \ln(2) \]

Using the property of logarithms that states \( \ln(e^x) = x \), we simplify the left side: \[ t = \ln(2) \]

The value of \( \ln(2) \) is approximately: \[ t \approx 0.6931 \]

Final Answer