Question - Solving Exponential Equation for x
Solution:
The equation provided in the image is:\[ e^{4-7x} + 11 = 20 \]To solve for $$ x $$, we'll 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 to get rid of the base $$ e $$.\[ \ln(e^{4-7x}) = \ln(9) \]3. Use the property of logarithms that $$ \ln(e^y) = y $$ to simplify the left side.\[ 4 - 7x = \ln(9) \]4. Subtract 4 from both sides to isolate the term with $$ x $$.\[ -7x = \ln(9) - 4 \]5. Finally, divide both sides by -7 to solve for $$ x $$.\[ x = \frac{\ln(9) - 4}{-7} \]6. Now you can either leave the answer in terms of natural logarithm or use a calculator to find a numerical value for $$ x $$.Using a calculator:\[ x \approx \frac{2.1972 - 4}{-7} \]\[ x \approx \frac{-1.8028}{-7} \]\[ x \approx 0.25754 \]So the solution to the equation is $$ x \approx 0.25754 $$.
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