Example Question - solving exponential equations
Here are examples of questions we've helped users solve.
Solving a System of Exponential Equations
The image shows a system of equations that needs to be solved: \(2^{x+y} = 8\) and \(2^{3x-y} = 128\) For the first equation, we can rewrite 8 as \(2^3\) since \(2^3 = 8\). This gives: \(2^{x+y} = 2^3\) From the property of exponents that states if \(a^m = a^n\) then \(m=n\) for any non-zero base \(a\), we get: \(x + y = 3\) (Equation 1) For the second equation, we can rewrite 128 as \(2^7\), since \(2^7 = 128\). This gives: \(2^{3x-y} = 2^7\) Using the same property of exponents, we get: \(3x - y = 7\) (Equation 2) Now we have a system of two linear equations in two variables (\(x\) and \(y\)): 1. \(x + y = 3\) 2. \(3x - y = 7\) We can solve this system by adding Equation 1 and Equation 2 together to eliminate \(y\): \(x + y + 3x - y = 3 + 7\) \(4x = 10\) \(x = 10/4\) \(x = 2.5\) With \(x\) found, we can substitute it back into Equation 1 to find \(y\): \(2.5 + y = 3\) \(y = 3 - 2.5\) \(y = 0.5\) Therefore, the solution to the system is \(x = 2.5\) and \(y = 0.5\).
Solving Exponential and Linear Equation Numerically
To solve the equation \(e^{x} - 7x + 11 = 20\), let's first simplify it by subtracting 20 from both sides to set it to zero: \[e^{x} - 7x + 11 - 20 = 0\] \[e^{x} - 7x - 9 = 0\] This equation cannot be solved algebraically due to the presence of both the exponential function and the linear term in x. We'll need to use numerical methods or graphing techniques to find an approximate solution. If you have access to graphing software, you could graph the function \(f(x) = e^{x} - 7x - 9\) and look for the x-values where the graph crosses the x-axis. Alternatively, you can use numerical methods such as Newton-Raphson, bisection, or other root-finding algorithms to find the solution to this equation. These methods are iterative and usually require a starting guess. For an initial guess, you can consider that for large values of x, the \(e^{x}\) term will dominate, so your solution is likely to be for a relatively small x value where the \( -7x \) term can balance the \(e^{x}\) and the constant -9. Would you like an approximation of the solution using a numerical method, or do you have a specific method in mind that you'd like to use?
Solving Exponential Equations with Natural Logarithms
To solve the equation provided in the image, we should isolate the exponential term on one side: \( e^{4-7x} + 11 = 20 \) Subtract 11 from both sides: \( e^{4-7x} = 20 - 11 \) \( e^{4-7x} = 9 \) Now take the natural logarithm (ln) of both sides to remove the base e: \( \ln(e^{4-7x}) = \ln(9) \) By the property of logarithms, \(\ln(e^{y}) = y\), so: \( 4 - 7x = \ln(9) \) Now to isolate x, we first subtract 4 from both sides: \( -7x = \ln(9) - 4 \) Finally, we divide by -7: \( x = \frac{\ln(9) - 4}{-7} \) \( x = \frac{-4 - \ln(9)}{7} \) This is the solution for x in terms of natural logarithms.
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 Equations with Natural Logarithms
To solve the equation \( e^{4-7x} + 11 = 20 \), follow these steps: 1. Subtract 11 from both sides of the equation: \[ e^{4-7x} + 11 - 11 = 20 - 11 \] \[ e^{4-7x} = 9 \] 2. Now take the natural logarithm (ln) of both sides to solve for \( 4-7x \): \[ \ln(e^{4-7x}) = \ln(9) \] Using the property of logarithms, \( \ln(e^y) = y \), we have: \[ 4 - 7x = \ln(9) \] 3. Solve for \( x \): \[ -7x = \ln(9) - 4 \] \[ x = \frac{-\ln(9) + 4}{7} \] Use a calculator to compute the value of \( \ln(9) \), and then compute \( x \).
Solving a Transcendental Equation
The equation in the image seems to state: \[ e^{x} + 7x + 11 = 20 \] To solve for \( x \), follow these steps: 1. First isolate the exponential term by moving all other terms to the other side of the equation. \[ e^{x} = 20 - 7x - 11 \] \[ e^{x} = 9 - 7x \] 2. This equation is not easily solvable using elementary algebra due to the presence of both an exponential term and a linear term in \( x \). You will have to use numerical methods, graphing, or an iterative technique like Newton-Raphson to find an approximate solution for \( x \). Unfortunately, a precise algebraic solution can't be found since we cannot isolate \( x \) with elementary algebraic operations. It is a transcendental equation. Do you want to proceed using a numerical method to estimate the solution?
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