Question - Solving Exponential Equation with Natural Logarithm
mathnatural logarithmsolving exponential equationexponential term isolationlogarithmic propertycalculating exponential equations
Solution:
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.
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