Problem 19 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(4^{x+2}=4^{2 x}\)

Step-by-Step Solution

Answer

The solution is \( x = 2 \). After checking, both sides of the equation are equal.

Step 1: Equalize the bases

Recognize that the equation already has the same base (4 on both sides). Thus, since the base is the same, we can set the exponents equal to each other: \[ x + 2 = 2x \]

Step 2: Solve for x

To isolate \( x \), we need to move all terms involving \( x \) to one side of the equation. Subtract \( x \) from both sides: \[ x + 2 - x = 2x - x \] Simplifying both sides gives us: \[ 2 = x \]

Step 3: Check the solution

Substitute \( x = 2 \) back into the original equation to ensure it holds true. When \( x = 2 \), the equation becomes: \[ 4^{2+2} = 4^{2 \times 2} \] This simplifies to: \[ 4^4 = 4^4 \] Since both sides equal \( 256 \), the solution \( x = 2 \) is confirmed as correct.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in mathematics, important for unraveling the unknown values in a problem. In the process of solving exponential equations, you aim to manipulate the equation such that you isolate the variable (often denoted as \(x\)). For example, let's consider the equation \(4^{x+2}=4^{2x}\). Since the bases are the same on both sides (base 4), the problem simplifies to solving the exponents: \(x + 2 = 2x\).
To solve for \(x\), arrange all terms involving \(x\) on one side of the equation. The term \(2x\) is already isolated on one side, so subtract \(x\) from both sides to simplify and isolate the \(x\) term. You'll end up with \(2 = x\).
This simple method of aligning and isolating terms is crucial in solving not just exponential equations but a wide array of algebraic problems.
By practicing the rearranging and solving of equations, students enhance their problem-solving abilities and grow more confident in handling mathematical challenges.

Equation Verification

Verifying your solution is a critical step in solving equations. It confirms whether the answer you found is indeed correct. For our example, when \(x = 2\), substitute it back into the original equation \(4^{x+2}=4^{2x}\) to see if both sides are equal.
Rewriting it with \(x = 2\) gives:

• Left Side: \(4^{2+2} = 4^4\)
• Right Side: \(4^{2 \times 2} = 4^4\)

Both sides simplify to \(4^4 = 256\). When both sides are equal, you confirm that \(x = 2\) is correct.
Verification helps catch any errors made in calculations or assumptions and solidifies understanding by ensuring the steps taken were correct. For anyone solving equations, this check is as vital as the solution process itself, providing confidence in your final answer.

Base Equalization

In exponential equations, having the same base on both sides simplifies the solving process. Base equalization means recognizing or transforming the equation such that both sides have identical bases. If already equal, as in \(4^{x+2} = 4^{2x}\), you can directly equate the exponents.
This technique, 'Base Equalization,' sometimes requires manipulating one or both sides of the equation. For instance, if you had \(2^{x+3} = 8^{x}\), you would first recognize that \(8\) can be rewritten as an exponential form of \(2\), specifically \(2^3\). Rewriting \(8^x\) as \((2^3)^x\), you simplify further to \(2^{3x}\). Then, your equation \(2^{x+3} = 2^{3x}\) allows for the exponents to be directly compared.
This method is particularly useful in ensuring equations are manageable and builds a strong foundation for understanding larger exponential and logarithmic expressions. Look for opportunities to equalize bases, as it leads to clearer, more straightforward solutions.