Example Question - exponential term isolation
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Solving Exponential Equations with Natural Logarithms
To solve the equation \( e^{4-7x} + 11 = 20 \), let's follow these steps: 1. Isolate the exponential term on one side of the equation: \( e^{4-7x} + 11 - 11 = 20 - 11 \) \( e^{4-7x} = 9 \) 2. To solve for \( x \), take the natural logarithm of both sides: \( \ln(e^{4-7x}) = \ln(9) \) Recall that \( \ln(e^y) = y \), so: \( 4 - 7x = \ln(9) \) 3. Solve for \( x \): \( -7x = \ln(9) - 4 \) \( x = \frac{-(\ln(9) - 4)}{7} \) \( x = \frac{4 - \ln(9)}{7} \) Thus, the solution to the equation is \( x = \frac{4 - \ln(9)}{7} \).
Solving Exponential Equation with Natural Logarithm
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.
Solving Exponential Equation with Natural Logarithm
The equation provided in the image is: e^(4 - 7x) + 11 = 20 To solve for x, follow these steps: 1. Subtract 11 from both sides of the equation to isolate the exponential term: e^(4 - 7x) = 20 - 11 e^(4 - 7x) = 9 2. 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) - 4 5. Divide both sides by -7 to solve for x: x = (ln(9) - 4) / -7 Using a calculator to find the numerical value of ln(9): x ≈ (2.1972 - 4) / -7 x ≈ (-1.8028) / -7 x ≈ 0.257543 So, the solution for x is approximately 0.2575 (rounded to four decimal places).
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