Question - Solving Exponential Equation with Natural Logarithm
Solution:
The equation provided in the image is:e^(4 - 7x) + 11 = 20To solve for x, follow these steps:1. Subtract 11 from both sides of the equation to isolate the exponential term:e^(4 - 7x) = 20 - 11e^(4 - 7x) = 92. Take the natural logarithm (ln) of both sides to eliminate the base e:ln(e^(4 - 7x)) = ln(9)3. Use the property of logarithms that says ln(a^b) = b*ln(a):(4 - 7x) * ln(e) = ln(9)Since ln(e) = 1, you can simplify this to:4 - 7x = ln(9)4. Isolate the term with x:-7x = ln(9) - 45. Divide both sides by -7 to solve for x:x = (ln(9) - 4) / -7Using a calculator to find the numerical value of ln(9):x ≈ (2.1972 - 4) / -7x ≈ (-1.8028) / -7x ≈ 0.257543So, the solution for x is approximately 0.2575 (rounded to four decimal places).
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