Problem 42 - A Graphical Approach to College Algebra

Question

Solve each equation. $$5^{2 x+1}=25$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{2}\).

Step 1: Recognizing Base and Exponent Relationship

Notice that 25 can be rewritten as a power of 5. Since 25 is equivalent to 5 squared, we can express it as \(5^2\). Thus, the given equation \(5^{2x+1} = 25\) can be rewritten as \(5^{2x+1} = 5^2\).

Step 2: Equalizing Exponents

Since the exponential terms have the same base, we can set their exponents equal to each other. Therefore, we have: \(2x + 1 = 2\).

Step 3: Solving the Equation for x

To isolate \(x\), first subtract 1 from both sides of the equation: \(2x + 1 - 1 = 2 - 1\), which simplifies to \(2x = 1\).

Step 4: Final Calculation

Divide both sides by 2 to solve for \(x\): \(x = \frac{1}{2}\).

Key Concepts

Base and Exponent Relationship

In exponential equations, the base and exponent relationship is crucial. The base is the number that is multiplied by itself repeatedly. The exponent signifies how many times the base is used as a factor. When an equation has the same base on both sides, it provides an effective way to simplify and solve equations.

Take the equation provided: \(5^{2x+1} = 25\). Here, the base is 5 for the left side of the equation. On the right side, 25 can be expressed using the same base: \(25 = 5^2\). Recognizing this relationship allows us to rewrite the equation to have the same base: \(5^{2x+1} = 5^2\). This clarity simplifies the equation significantly.

Solving Equations

Once we establish a common base, solving the equation becomes straightforward. We focus on the powers and set the exponents equal because the bases are already the same. For our equation, this means setting \(2x + 1 = 2\). This step is crucial because it turns a seemingly complex exponential equation into a simple linear one.

From here, we solve for \(x\) using basic algebra. First, subtract 1 from both sides to isolate the term with \(x\):

• \(2x + 1 - 1 = 2 - 1\)
• This simplifies to \(2x = 1\).

By equalizing the exponents, we streamline the process of finding the unknown variable \(x\).

Exponentiation

Exponentiation involves raising a base to a certain power, a fundamental concept in mathematics. When the base and exponent are involved in a relationship, like in our equation, the exponent dictates how many copies of the base are multiplied together.

For example, \(5^3\) means \(5\times5\times5\), which equals 125. Understanding exponentiation helps in rewriting and simplifying equations. The key is recognizing patterns, such as how any number raised to the power of one is itself, or how zero as an exponent will result in one if the base is nonzero.

This understanding is essential when manipulating equations to achieve common bases, as shown by re-expressing 25 as \(5^2\). It lays the groundwork for solving more complex exponential problems efficiently.