Problem 70 - Elementary and Intermediate Algebra

Question

Solve each equation. Give the exact solution and, when appropriate, an approximation to four decimal places. $$ 5^{x+4}=125 $$

Step-by-Step Solution

Answer

The exact solution is \(x = -1\). No approximation needed.

Step 1: Recognize the Base Relationship

Notice that 125 can be expressed as a power of 5. Specifically, 125 is equal to \(5^3\). This base relationship tells us the exponent needed to express 125 as a power of 5.

Step 2: Set Up the Equation

Substitute \(125\) with \(5^3\) in the equation. This transforms the original equation to: \[5^{x+4} = 5^3\] Because the bases are the same, we can equate the exponents.

Step 3: Solve for x by Equating Exponents

Set the exponents equal to each other: \(x+4 = 3\). To isolate \(x\), subtract 4 from both sides of the equation, giving us: \(x = 3 - 4\).

Key Concepts

Base Relationship

Understanding the base relationship is key when working with exponential equations. In mathematics, the base of an exponent is the number that is being multiplied by itself. Recognizing the connection between bases can simplify solving these equations.
For example, when you see an equation like \( 5^{x+4} = 125 \), you need to determine if 125 can be expressed as a power of 5. It turns out that 125 is the same as \( 5^3 \). Recognizing this allows us to rewrite the equation with matching bases. This step is crucial because having the same base on both sides of the equation lets us focus on the exponents to find a solution.
In summary, recognizing equivalent base powers can convert a complex problem into one that is easier to solve.

Exponentiation

Exponentiation is about raising a base to a certain power, and it's a common operation in algebra. When we talk about a number like \( 5^{x+4} \), the number 5 is the base, and \( x+4 \) is the exponent. The expression means that the base is multiplied by itself the number of times indicated by the exponent.
For example, in the equation \( 5^{x+4} = 5^3 \), you see the power relationship at work. Here, both sides have the base 5, but they are raised to different powers. Exponentiation simplifies the expression of large numbers because you can express huge values compactly. It also provides a way to extend numbers into very large or very small realms, as seen with scientific notation.
When bases are identical, you can equate the exponents directly, as shown when transforming \( 5^{x+4} = 125 \) into \( 5^{x+4} = 5^3 \). This creates a simpler algebraic problem to solve.

Isolating Variables

To isolate a variable means to get the variable by itself on one side of the equation. This step is often necessary to solve for the unknown. In exponential equations where you've aligned the bases, isolating the variable becomes a straightforward task.
In the example, once you've equated the exponents from \( 5^{x+4} = 5^3 \), you have \( x+4 = 3 \). Solving this requires basic algebraic manipulation. You need to subtract 4 from both sides of the equation to isolate \( x \):

• Original exponent equation: \( x+4 = 3 \)
• Subtracting 4 from both sides: \( x+4-4 = 3-4 \)
• The simplified result: \( x = -1 \)

In this way, isolating variables helps solve for the unknowns in equations, allowing us to determine the exact values that make the equation true.