Example Question - solve exponential equation
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Solving Exponential Equation
The given equation to solve is: e^(4 - 7x) + 11 = 20 First, isolate the exponential term by subtracting 11 from both sides: e^(4 - 7x) + 11 - 11 = 20 - 11 e^(4 - 7x) = 9 Now, to solve for x, apply the natural logarithm (ln) to both sides, because ln(e^y) = y: ln(e^(4 - 7x)) = ln(9) This simplifies to: 4 - 7x = ln(9) Now, isolate x by subtracting 4 from both sides, and then divide by -7: 4 - 7x - 4 = ln(9) - 4 -7x = ln(9) - 4 x = (ln(9) - 4) / (-7) Using a calculator to find the numerical approximation for ln(9): x ≈ (2.1972 - 4) / (-7) x ≈ (-1.8028) / (-7) x ≈ 0.25754 Therefore, the solution to the equation is approximately x ≈ 0.25754.
Solving Exponential Equation
The image shows the following exponential equation: e^(4 - 7x) + 11 = 20 To solve the equation for x, we'll follow these steps: 1. First, isolate the exponential term: e^(4 - 7x) = 20 - 11 e^(4 - 7x) = 9 2. Next, we would take the natural logarithm (ln) of both sides to get the exponent by itself: ln(e^(4 - 7x)) = ln(9) 3. Since ln(e^y) = y for any y (because ln and e are inverse functions), we can simplify the left-hand side: 4 - 7x = ln(9) 4. Lastly, we'll solve for x: 7x = 4 - ln(9) x = (4 - ln(9)) / 7 To find the numerical value, compute: ln(9) ≈ 2.1972 So, x ≈ (4 - 2.1972) / 7 x ≈ 1.8028 / 7 x ≈ 0.2575 Therefore, the solution to the equation is x ≈ 0.2575.
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