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Chapter 3: Problem 58

Use the One-to-One Property to solve the equation for \(x\). $$e^{x^{2}+6}=e^{5 x}$$

Short Answer

Expert verified
The solutions for the given equation are \(x = 2\) and \(x = 3\).

Step by step solution

01

Apply the One-to-One Property of Exponential Functions

Make use of the one-to-one property of exponential functions. This property states that if two exponential expressions with the same base are equal, then their exponents must also be equal. As the equation \(e^{x^{2}+6} = e^{5x}\) has similar bases (base \(e\)) on both sides, we can equate the exponents: \(x^{2}+6 = 5x\).
02

Rearrange the equation to a standard Quadratic form

Rearranging the equation to a standard quadratic form involves putting all terms on one side of the equation and setting the equation to zero. To do this, subtract \(5x\) from both sides of the equation to get it in standard quadratic form: \(x^{2}-5x+6 = 0\).
03

Solve the Quadratic Equation

The quadratic equation is in the form \(ax^{2} + bx + c = 0\). We can solve the equation \(x^{2}-5x+6 = 0\) by factoring. The factors of +6 that sum to -5 are -2 and -3. Hence, we can write the expression as: \((x - 2)(x - 3) = 0\).
04

Solve for x

Set each factor equal to zero and solve the two resulting equations to find the values of \(x\). Thus, we have: \(x - 2 = 0\) gives \(x = 2\), and \(x - 3 = 0\) gives \(x = 3\). These are the solutions for the equation.

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Most popular questions from this chapter

The populations \(P\) (in thousands) of Pittsburgh, Pennsylvania from 2000 through 2007 can be modeled by \(P=\frac{2632}{1+0.083 e^{0.0500 t}}\) where \(t\) represents the year, with \(t=0\) corresponding to \(2000 .\) (Source: U.S. Census Bureau) (a) Use the model to find the populations of Pittsburgh in the years \(2000,2005,\) and 2007 . (b) Use a graphing utility to graph the function. (c) Use the graph to determine the year in which the population will reach 2.2 million. (d) Confirm your answer to part (c) algebraically.

Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 4 \\ \hline y & 5 & 1 \\ \hline \end{array} $$

A logarithmic model has the form ________ or ________.

The graph of a Gaussian model is ________ shaped, where the ________ ________ is the maximum -value of the graph.

The populations \(P\) (in thousands) of Horry County, South Carolina from 1970 through 2007 can be modeled by \(P=-18.5+92.2 e^{0.0282 t}\) where \(t\) represents the year, with \(t=0\) corresponding to 1970\. (Source: U.S. Census Bureau) (a) Use the model to complete the table. $$ \begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1970 & 1980 & 1990 & 2000 & 2007 \\ \hline \text { Population } & & & & & \\ \hline \end{array} $$ (b) According to the model, when will the population of Horry County reach \(300,000 ?\) (c) Do you think the model is valid for long-term predictions of the population? Explain.
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