Question - Solving an Exponential Equation
Solution:
This is an equation involving an exponential function. To solve the equation $$e^{4 - 7x} + 11 = 20$$, we need to isolate the term with the exponent and then solve for $$x$$.Here are the steps to solve the equation:1. Subtract 11 from both sides to isolate the exponential term: $$e^{4 - 7x} = 9$$2. Take the natural logarithm (ln) of both sides to remove the base $$e$$ and get the exponent by itself: $$ \ln(e^{4 - 7x}) = \ln(9) $$3. The natural logarithm and the exponential function are inverse functions, so: $$4 - 7x = \ln(9)$$4. Solve for $$x$$: $$7x = 4 - \ln(9)$$ $$x = \dfrac{4 - \ln(9)}{7}$$5. To find the numeric value, we can use a calculator: The natural logarithm of 9 (to two decimal places) is approximately 2.20. $$x \approx \dfrac{4 - 2.20}{7}$$ $$x \approx \dfrac{1.80}{7}$$ $$x \approx 0.2571$$So the approximate solution to the original equation is $$x \approx 0.2571$$.
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