Problem 142 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

Step-by-Step Solution

Answer

The first two equations \(10^{x}=5.71\) and \(e^{x}=0.72\) are exponential equations, while the third equation \(x^{10}=5.71\) is not an exponential equation. The corrected version of the third equation could be \(a^{x}=5.71\), where 'a' is a constant.

Step 1 Analyzing Equation 1

The first equation is \(10^{x}=5.71\). An exponential equation is in the form \(a^{x} = b\), where 'a' is a constant, 'x' is the variable in the exponent and 'b' is the result. In the first equation, \(10^{x} = 5.71\), '10' is the constant, 'x' is the variable in the exponent and '5.71' is the result. Therefore, the first equation \(10^{x} = 5.71\) is an exponential equation.

Step 2 Analyzing Equation 2

The second equation is \(e^{x}=0.72\). In this equation, 'e' is the constant, 'x' is the variable in the exponent and '0.72' is the result. Therefore, the second equation \(e^{x} = 0.72\) is an also an exponential equation.

Step 3 Analyzing Equation 3

The third equation is \(x^{10}=5.71\). In this equation, 'x' is the variable, '10' is constant exponent and '5.71' is the result. Since the constant is in the exponent and the variable is the base, this equation does not qualify as an exponential equation. To make it correct, we need to correct the base and the exponent so the base becomes a constant and the exponent becomes a variable, i.e., \(a^{x}=5.71\) where 'a' is a constant value.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. In algebraic terms, these equations often involve variables, constants, and algebraic operations such as addition, subtraction, multiplication, and division.
These equations come in many forms, including linear, quadratic, polynomial, and exponential.
- **Linear equations**: These are algebraic equations where each term is a constant or the product of a constant and a single variable. - **Quadratic equations**: These typically involve variables raised to the power of two. - **Polynomial equations**: These involve variables raised to various powers, more than two.
In exponential equations, the variable appears in the exponent, distinguishing them from other types like polynomials. Understanding these forms helps to identify and solve them correctly in algebra.

Equation Analysis

Equation analysis involves understanding the components of an equation to determine its type and properties. This is a crucial step in solving mathematical problems.

To analyze an equation, you should:

• Identify the **constant** (a fixed value) in the equation.
• Determine the **variable**, which is the unknown part of the equation that might change.
• Check the location of these components; if the variable is in the exponent, it's exponential.

In the exercise, we looked closely at equations like \(10^x = 5.71\) and \(e^x = 0.72\). Here, the exponent holds the variable, marking them as their respective exponential equations. On the other hand, \(x^{10} = 5.71\) isn't exponential because the variable isn't in the exponent.

Mathematical Constants

Mathematical constants are unique numbers with inherent values that do not change. They serve crucial roles in various equations and formulas.

Some well-known constants in mathematics include:

• \(e\) (Euler’s number): Approximately 2.718, appearing frequently in exponential growth and decay problems.
• \(\pi\) (pi): Approximately 3.14159, used abundantly in calculations involving circles.

In exponential equations, constants serve as the base of the power. For instance, in \(e^x = 0.72\), \(e\) is the constant base.
Recognizing these constants is vital for understanding and solving complex equations efficiently.