Example Question - solving exponential equation
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Solving Exponential Equation with Identical Base
문제는 다음과 같이 주어집니다: \(3^{-1-\sqrt5} \times 3^{1+\sqrt5}\) 이 문제를 해결하는 과정은 다음과 같습니다: 우선 지수법칙을 사용하여 같은 밑을 가진 지수의 곱셈을 처리합니다: \(a^m \times a^n = a^{m+n}\) 위의 법칙을 적용합시다: \(3^{-1-\sqrt5} \times 3^{1+\sqrt5} = 3^{(-1-\sqrt5) + (1+\sqrt5)} = 3^{0}\) 어떠한 수를 0으로 거듭제곱하면 항상 1입니다. 즉: \(3^{0} = 1\) 따라서 주어진 식의 값은 1입니다. 선택지를 확인하니 1에 해당하는 선택지는 (3)번입니다. 따라서 올바른 답은 (3) 1입니다.
Solving Exponential Equation with Logarithms
The equation given in the image is: 5^(3-x) + 2 = 8 To solve for x, we should first isolate the exponential term on one side of the equation: 5^(3-x) + 2 - 2 = 8 - 2 5^(3-x) = 6 Now, to solve for x, we need to rewrite the equation in logarithmic form. Taking the logarithm of both sides (using a common logarithm, base 10, or a natural logarithm, base e), we can apply the properties of logarithms to solve for the exponent. But first, let's use the natural logarithm, as it's often the one used in these types of problems: ln(5^(3-x)) = ln(6) Applying the property of logarithms that allows us to bring down the exponent, we get: (3 - x)ln(5) = ln(6) Now, divide both sides by ln(5) to isolate (3 - x): 3 - x = ln(6) / ln(5) Now, solve for x: x = 3 - ln(6) / ln(5) Using a calculator to find the values of ln(6) and ln(5) and then performing the subtraction: x ≈ 3 - (1.79176 / 1.60944) x ≈ 3 - 1.1134 x ≈ 1.8866 Rounded to the nearest thousandth, the value of x is approximately: x ≈ 1.887
Solving an Equation with Exponential and Linear Terms
The equation given in the image is: \[ e^{4x} - 7x + 11 = 20 \] To solve for \( x \), we want to isolate \( x \) on one side. However, because the equation includes both an exponential and a linear term in \( x \), it cannot be solved using elementary algebraic methods. This equation would typically require numerical methods to solve, such as Newton-Raphson, or graphing techniques to find an approximate solution. To begin, let's simplify the equation by subtracting 20 from both sides: \[ e^{4x} - 7x + 11 - 20 = 0 \] \[ e^{4x} - 7x - 9 = 0 \] Now you would typically use a computational tool or graphing calculator to find the roots of this transcendental equation. Unfortunately, without such tools at my disposal, I cannot compute the exact solution, but you can certainly employ these methods to find a numerical solution for \( x \).
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 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: \( e^{4 - 7x} + 11 = 20 \) 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 for Unknown Variable
The equation in the image is \( e^{x - 7} + 11 = 20 \). To solve for \( x \), we want to isolate the term containing \( x \). Here are the steps: 1. Subtract 11 from both sides of the equation to get the exponential term by itself: \( e^{x - 7} = 20 - 11 \) \( e^{x - 7} = 9 \) 2. Take the natural logarithm (ln) of both sides to 'undo' the exponential. Since \( \ln(e^y) = y \), we have: \( \ln(e^{x - 7}) = \ln(9) \) \( x - 7 = \ln(9) \) 3. Add 7 to both sides to solve for \( x \): \( x = \ln(9) + 7 \) Now we have the solution for \( x \). If a specific numerical value is required, you'd compute \( \ln(9) \) using a calculator and then add 7 to that result.
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: \( x \approx \ln(9) \approx 2.1972 \)
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).
Solving Exponential Equation
This is an exponential equation. Let's solve it step by step: The equation is `e^(4-7x) + 11 = 20`. First, subtract 11 from both sides to isolate the exponential term: `e^(4-7x) = 9`. Next, we'll take the natural logarithm (ln) of both sides to get rid of the exponential expression and solve for x: `ln(e^(4-7x)) = ln(9)`. Using the property of logarithms: `4 - 7x = ln(9)`. Now, we just need to solve for x: `-7x = ln(9) - 4`. Divide both sides by -7 to solve for x: `x = (ln(9) - 4) / -7`. This gives you the value of x in terms of the natural logarithm of 9. Use a calculator to find a numerical value if required.
Solving Exponential Equation
The equation provided in the image is: \( e^{4x} - 7x + 11 = 20 \) To solve for \( x \), follow these steps: 1. Isolate the exponential term by moving all other terms to the other side of the equation: \( e^{4x} = 20 - 11 \) 2. Simplify the right side: \( e^{4x} = 9 \) 3. Take the natural logarithm of both sides to remove the exponential: \( \ln(e^{4x}) = \ln(9) \) 4. Knowing that \( \ln(e^y) = y \), simplify the left side: \( 4x = \ln(9) \) 5. Finally, solve for \( x \) by dividing both sides by 4: \( x = \frac{\ln(9)}{4} \) Therefore, the solution to the equation is: \( x = \frac{\ln(9)}{4} \)
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