Solving Exponential Equations with Natural Logarithms
To solve the equation \( e^{4-7x} + 11 = 20 \), follow these steps:
1. Subtract 11 from both sides of the equation:
\[ e^{4-7x} + 11 - 11 = 20 - 11 \]
\[ e^{4-7x} = 9 \]
2. Now take the natural logarithm (ln) of both sides to solve for \( 4-7x \):
\[ \ln(e^{4-7x}) = \ln(9) \]
Using the property of logarithms, \( \ln(e^y) = y \), we have:
\[ 4 - 7x = \ln(9) \]
3. Solve for \( x \):
\[ -7x = \ln(9) - 4 \]
\[ x = \frac{-\ln(9) + 4}{7} \]
Use a calculator to compute the value of \( \ln(9) \), and then compute \( x \).
