Example Question - isolating exponential term
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Solving an Exponential Equation
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 \(x \approx 0.2571\).
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.
Solving Exponential Equation for x
The equation in the image is: e^(4 - 7x) + 11 = 20 To solve for x, follow these steps: 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 exponent: ln(e^(4 - 7x)) = ln(9) 3. Since ln(e^y) = y for any y, this simplifies to: 4 - 7x = ln(9) 4. Now isolate x by first subtracting 4 from both sides: -7x = ln(9) - 4 5. Finally, divide both sides by -7 to solve for x: x = (ln(9) - 4) / -7 This will give you the numeric value of x. To find the exact value you can use a calculator.
Solving Exponential Equation with Natural Logarithm
The equation in the image is \( e^x + 11 = 20 \). To solve for \( x \), you want to isolate \( e^x \) on one side of the equation. Here are the steps to do that: 1. Subtract 11 from both sides of the equation to isolate the exponential term on one side: \( e^x + 11 - 11 = 20 - 11 \) Simplifying that, you get: \( e^x = 9 \) 2. Taking the natural logarithm (ln) of both sides to solve for \( x \): \( \ln(e^x) = \ln(9) \) Since \( \ln(e^x) \) simplifies to \( x \)(because \( \ln(e^x) \) and \( e^x \) are inverse functions), you have: \( x = \ln(9) \) The approximate value of \( x \) can be found using a calculator: \( x \approx \ln(9) \approx 2.1972 \)
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