Problem 1 - Elementary and 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 equation.

Step 1: Identify the Equation Type

The given equation is of the form \(3^{2x} = 8\). It has a constant base \(3\) and a variable \(2x\) in its exponent. This form is characteristic of a specific type of equation.

Step 2: Recognize the Equation Characteristics

In the equation \(3^{2x} = 8\), the unknown variable \(x\) appears as the exponent. These types of equations are known as exponential equations because the variable is in the exponent.

Step 3: Fill in the Blank with the Correct Term

Based on the identification of an equation with a variable in the exponent, the appropriate term to fill in the blank is 'exponential'. Thus, the equation \(3^{2x} = 8\) is an exponential equation.

Key Concepts

Constant Base

Exponential equations often start with something called a *constant base*. This means the base of the exponent is a fixed number, like 2, 3, or 5 - numbers that remain the same in the problem. They don't change when solving the equation. Imagine you have a mountain that never moves, while the height can grow or shrink. The mountain represents the constant base in exponential equations.For example, in the equation \(3^{2x} = 8\), the base of 3 doesn't change as you try to solve for \(x\). This is why it's called a constant base - it stays steady no matter the height of the tower (or power) it's supporting.

Variable Exponent

In exponential equations, the exponent is where things get exciting. Here, we have what's known as a *variable exponent*. Instead of being a fixed number, the exponent is something we need to figure out. Think of the exponent like a mystery to solve. It's what you're looking for, just like a detective. The variable - often \(x\) - needs to be found because it's the unknown in the equation, as in \(3^{2x} = 8\).In simple terms, the exponent can twist and turn so the equation balances out. Unlike the constant base that stays solid, the variable exponent will be your target, allowing you to manipulate and solve for whatever value makes the equation true.

Solving Exponential Equations

Solving exponential equations can be an adventure, but let's make it simple. When you have an equation like \(3^{2x} = 8\), your goal is to figure out what \(x\) is.- **Step One:** Recognize the equation's form. Notice you have a constant base and a variable exponent.- **Step Two:** Use mathematical tools, like logarithms or making both sides have the same base, to find \(x\).- **Step Three:** Check your solution to ensure it's correct. Substitute \(x\) back into the original equation to see if both sides match.Start by relabeling the equation, so you understand what you're working with. Then, apply some clever math strategies to find that hidden value in the exponent. It may seem tricky at first, but breaking down each part will help you discover the solution step-by-step.