Solving Exponential Equation for x
The equation provided is:
\[ e^{4 - 7x} + 11 = 20 \]
To solve for \( x \), follow these steps:
1. Subtract 11 from both sides of the equation:
\[ e^{4 - 7x} = 9 \]
2. Take the natural logarithm (ln) of both sides:
\[ \ln(e^{4 - 7x}) = \ln(9) \]
3. Because the natural logarithm and the exponential function are inverse operations, \( \ln(e^{y}) = y \). Hence:
\[ 4 - 7x = \ln(9) \]
4. To isolate \( x \), subtract 4 from both sides:
\[ -7x = \ln(9) - 4 \]
5. Finally, divide both sides by -7 to solve for \( x \):
\[ x = \frac{\ln(9) - 4}{-7} \]
So,
\[ x = \frac{4 - \ln(9)}{7} \]
Now you can use a calculator to find the approximate value of \( x \).
