Problem 144 - College Algebra

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 equation \(10^{x}-5.71\) and the second equation \(e^{x}-0.72\) are both exponential equations. The third equation \(x^{10}-5.71\) is not an exponential equation; it is instead a polynomial equation. We can amend it to become exponential by changing it to \(10^{x}-5.71\).

Step 1: Determine nature of first equation

Look at the equation \(10^{x}-5.71\). Notice how \(x\) is the exponent, which is the hallmark of an exponential equation. Therefore, the first equation \(10^{x}-5.71\) is an exponential equation.

Step 2: Determine nature of second equation

Inspect the equation \(e^{x}-0.72\). Similar to the first equation, \(x\) is in the position of the exponent, hence, the equation \(e^{x}-0.72\) is an exponential equation as well.

Step 3: Determine nature of third equation

Review the equation \(x^{10}-5.71\). Here, the variable \(x\) is the base, not the exponent, making it a polynomial equation and not an exponential equation. To make it exponential, we can adjust it to be \(10^{x}-5.71\)

Key Concepts

Understanding Polynomial Equations

Polynomial equations are mathematical expressions formed by summing various terms of the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n, a_{n-1}, \ldots, a_0\) are constants, and \(n, n-1, \ldots, 1, 0\) are non-negative integer exponents. In simple terms, these are equations involving variables raised to whole number powers. Unlike exponential equations, in polynomials, the variable is the base, not the exponent. This distinguishes polynomial equations from other types of equations.

Key characteristics of polynomial equations include:

• They can have multiple terms, but each term's exponent of the variable is a non-negative integer.
• They help in understanding properties such as roots, factorization, and the behavior of graphs.
• A polynomial equation of degree 2 is known as a quadratic equation, degree 3 is cubic, and so on.

Understanding when an equation is a polynomial versus another type, such as exponential, is critical in the study of mathematics.

Getting to Know Exponents

Exponents are mathematical notations indicating the number of times a number, known as the base, should be multiplied by itself. An expression like \(x^n\) means that \(x\) is multiplied by itself \(n\) times.

Here are some key aspects of exponents:

• When a positive integer is used as an exponent, it signifies repeated multiplication.
• The exponent of zero always results in one, i.e., \(x^0 = 1\), provided \(x eq 0\).
• Negative exponents indicate reciprocal multiplication, such as \(x^{-n} = \frac{1}{x^n}\).

Exponents provide a simple and clear way to represent larger numbers compactly and are fundamental in exponential equations where the variable serves as the exponent. Understanding exponents is crucial for working with both polynomial and exponential equations.

Approaching Equation Analysis

Equation analysis involves examining the structure of an equation to understand its type and behavior. This is essential in effectively solving them. When analyzing an equation:

• Identify whether it is a polynomial, exponential, or another form of equation.
• Consider characteristics like the arrangement of terms and the position of variables or constants.
• Use tools like factoring, graphing, or substitution to simplify or solve the equation.

In the original exercise, analyzing the equations revealed their nature by the placement of the variable. This analysis helps decide the correct approach for solutions and adjustments. For instance, recognizing that \(x^{10} - 5.71\) differs from \(10^x - 5.71\) by the role of \(x\) prepares you to apply different mathematical strategies. Mastery of equation analysis significantly aids in higher-level math tasks.