Problem 1 - Intermediate Algebra

Question

Fill in the blanks. An equation with a positive constant base and a variable in its exponent, such as \(3^{2 x}=8,\) is called an ___ equation.

Step-by-Step Solution

Answer

Exponential

Step 1: Identify the Type of Equation

The equation given is \(3^{2x} = 8\). Notice that the variable \(x\) is in the exponent. Equations where the variable is in the exponent are called exponential equations.

Step 2: Understand the Characteristics of the Equation

An exponential equation is one in which a constant base is raised to a power that includes a variable. In this equation, the base is 3, and the exponent is \(2x\). These kinds of equations are used to model exponential growth or decay.

Step 3: Fill in the Blank

Based on the identification in Step 1, fill in the blank in the statement. The blank requires the term for this type of equation, which is "exponential."

Key Concepts

Exponent

The exponent in an equation dictates how many times the base is multiplied by itself. For example, in the equation \(3^{2x}\), the exponent is \(2x\). Exponents can be numbers, but in exponential equations, they often include variables to represent unknowns. Exponents are powerful tools in mathematics, allowing expressions of very large or small numbers compactly.

• They show how many times the base is used as a factor.
• When dealing with variables in exponents, it becomes an exponential equation.

Exponents are key in expressing exponential growth, where a quantity increases rapidly.

Variable

A variable represents an unknown or changeable number. In exponential equations, variables often appear in the exponent, such as in \(3^{2x}\). This is what makes the equation exponential. Variables allow for flexibility and modeling real-world scenarios.

• Variables can change values, allowing for multiple solutions to an equation.
• They are often denoted by letters, with \(x\) being commonly used.

When a variable is an exponent, the equation's solutions require solving for that variable, often involving logarithms.

Constant Base

The constant base in an exponential equation is the number that is raised to a power. In our example \(3^{2x} = 8\), the base is 3. Constant bases play a crucial role, as they determine how the equation behaves.

• A constant base provides stability, as it does not change with variables.
• It is a fixed number, setting the backdrop for the variable exponent.

Understanding the base is important for prediction in models that utilize exponential growth or decay.

Exponential Growth

Exponential growth occurs when a quantity increases by a consistent factor over equal increments, often modeled by exponential equations.In the equation \(3^{2x}\), the growth is determined by the changing variable in the exponent.

• Exponential growth starts slow but accelerates rapidly.
• It is common in areas like population growth and finance.

Recognizing exponential growth is critical in fields that project future scenarios, as it enables calculations involving exponentially increasing quantities.