Question - Solving Exponential Equation with Natural Logarithm
Solution:
To solve the equation $$ e^{4 - 7x} + 11 = 20 $$, you first isolate the exponential part:1. Subtract 11 from both sides to move the constant term to the right-hand side of the equation:\[ e^{4 - 7x} = 20 - 11 \]\[ e^{4 - 7x} = 9 \]2. Now take the natural logarithm (ln) of both sides to solve for $$ 4 - 7x $$:\[ \ln(e^{4 - 7x}) = \ln(9) \]Since the natural logarithm and the exponential function are inverse functions, $$ \ln(e^{y}) = y $$, you get:\[ 4 - 7x = \ln(9) \]3. Finally, solve for x:\[ 7x = 4 - \ln(9) \]\[ x = \frac{4 - \ln(9)}{7} \]This is the solution to the equation.
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