Problem 34
Question
Solve the exponential equation. Round to three decimal places, when needed. $$e^{x}+e^{-x}=-x+4$$
Step-by-Step Solution
Verified Answer
After applying the mentioned steps, you will get the two solutions for x. Plug these solutions into the original equation to verify the solution for x. Due to the complexity of this equation, numerical methods such as the Newton-Raphson method may be employed if it is difficult to solve algebraically.
1Step 1: Combine Like Terms
Firstly, the equation needs to rearrange in such a way that the left-hand side (LHS) and the right-hand side (RHS) contains similar terms. The LHS represents the exponential terms, and the RHS represents the linear term. Rewrite the original equation \(e^{x} + e^{-x} = -x + 4\) to \(-x + e^{x} + e^{-x} -4 = 0\)
2Step 2: Substitution
Consider \(y = e^{x}\), then \(1/y = e^{-x}\). The equation becomes \(-x + y + (1/y) -4 = 0\). Further solve this equation to \(xy^2+x-y^2-4y=0\). This equation is a quadratic equation in y.
3Step 3: Solve Quadratic Equation
Use the Quadratic Formula to solve for y: \(y = [-b \pm \sqrt{(b^2 - 4ac)}] / (2a)\), where a = x, b = -1, c = -4x in the equation. Solving with these values gives tow solutions for y which are values of \(e^{x}\).
4Step 4: Solve for x
As both solutions for y are values of \(e^{x}\), take the natural logarithm (ln) of both solutions to determine the corresponding x-values. It's important to use ln because it's the inverse function of the exponential function.
Key Concepts
Quadratic EquationNatural LogarithmExponential Function
Quadratic Equation
A quadratic equation is a fundamental concept in algebra, often seen in the form \(ax^2 + bx + c = 0\). This type of equation involves polynomials where the highest degree of the variable is 2. In this exercise, after substituting \(y = e^x\), we transform the exponential terms into a quadratic form.
In the process, the equation \(-x + y + \frac{1}{y} - 4 = 0\) becomes quadratic in terms of the variable \(y\). By simple mathematical manipulations, it takes the shape \(xy^2 + x - y^2 - 4y = 0\).
Quadratic equations can be solved by several methods, such as:
Understanding how to manipulate quadratic equations is key not only in solving algebra problems but also in comprehending more complex mathematical concepts.
In the process, the equation \(-x + y + \frac{1}{y} - 4 = 0\) becomes quadratic in terms of the variable \(y\). By simple mathematical manipulations, it takes the shape \(xy^2 + x - y^2 - 4y = 0\).
Quadratic equations can be solved by several methods, such as:
- Factoring
- Completing the Square
- Using the Quadratic Formula
Understanding how to manipulate quadratic equations is key not only in solving algebra problems but also in comprehending more complex mathematical concepts.
Natural Logarithm
The natural logarithm function, denoted as \(\ln(x)\), is another crucial concept when handling exponential equations. This special logarithm has a base \(e\), where \(e ≈ 2.718\), otherwise known as Euler's number.
Logarithms are hugely beneficial in mathematics as they serve as the inverse operation to exponentiation. This means if \(y = e^x\), then \(x = \ln(y)\). Thus, in the given problem, once determined the values for \(y\), taking the natural logarithm helps solve for \(x\).
Some important properties of natural logarithms include:
Logarithms are hugely beneficial in mathematics as they serve as the inverse operation to exponentiation. This means if \(y = e^x\), then \(x = \ln(y)\). Thus, in the given problem, once determined the values for \(y\), taking the natural logarithm helps solve for \(x\).
Some important properties of natural logarithms include:
- \(\ln(1) = 0\) because \(e^0 = 1\)
- \(\ln(e) = 1\) because \(e^1 = e\)
- \(\ln(ab) = \ln(a) + \ln(b)\)
- \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\)
Exponential Function
An exponential function is any mathematical function of the form \(f(x) = a \cdot b^x\), albeit in this particular exercise, the base is Euler's number \(e\), making the function \(f(x) = e^x\).
Exponential functions are fundamental in math because of their constant growth (or decay) rates. The constant 'e' makes these functions especially important in natural processes, such as population growth and radioactive decay, where changes occur continuously.
Key properties of exponential functions include:
Exponential functions are fundamental in math because of their constant growth (or decay) rates. The constant 'e' makes these functions especially important in natural processes, such as population growth and radioactive decay, where changes occur continuously.
Key properties of exponential functions include:
- \(e^0 = 1\)
- If \(a < 0\), \(e^a\) represents a decay function
- If \(a > 0\), \(e^a\) denotes growth
- The inverse of an exponential function is the logarithmic function
Other exercises in this chapter
Problem 33
Evaluate each expression without using a calculator. $$\log _{4} 4^{x^{2}+1}$$
View solution Problem 33
Sketch the graph of each function. $$f(x)=-4(3)^{x}+1$$
View solution Problem 34
Due to inflation, a dollar in the year 1994 is worth \(\$ 1.28\) in 2005 dollars. Find an exponential function \(v(t)=C e^{k t}\) that models the value of a 199
View solution Problem 34
State whether each function is one-to-one. $$f(x)=-\frac{1}{3} x^{3}-5$$
View solution