Example Question - calculating exponential equations

Solving Exponential Equation with Natural Logarithm

The equation in the image is: \( e^{4 - 7x} + 11 = 20 \) To solve for \( x \), follow these steps: 1. Isolate the exponential term by subtracting 11 from both sides of the equation: \( e^{4 - 7x} = 9 \) 2. Take the natural logarithm of both sides to eliminate the base \( e \): \( \ln( e^{4 - 7x} ) = \ln(9) \) 3. Use the logarithmic property \( \ln(e^y) = y \) to simplify the left side: \( 4 - 7x = \ln(9) \) 4. Isolate \( x \) by subtracting 4 from both sides and then dividing by -7: \( - 7x = \ln(9) - 4 \) \( x = \frac{\ln(9) - 4}{-7} \) 5. Calculate the value of \( x \): \( x \approx \frac{2.1972 - 4}{-7} \) \( x \approx \frac{-1.8028}{-7} \) \( x \approx 0.2575 \) Therefore, the solution for \( x \) is approximately 0.2575.