Problem 23 - Algebra 2 and Trigonometry

Question

Solve each equation and check. \(25^{x}=5^{x+3}\)

Step-by-Step Solution

Answer

The solution is \(x = 3\).

Step 1: Express Each Side with the Same Base

Notice that the equation has different bases. Express both sides with base 5: \[ 25^x = (5^2)^x = 5^{2x} \] So the equation becomes: \[ 5^{2x} = 5^{x+3} \]

Step 2: Set Exponents Equal

Since the bases are the same, set the exponents equal to each other: \[ 2x = x + 3 \]

Step 3: Solve for x

Isolate the variable \(x\) by subtracting \(x\) from both sides of the equation: \[ 2x - x = 3 \] Simplify: \[ x = 3 \]

Step 4: Check the Solution

Substitute \( x = 3 \) back into the original equation to verify: - Check the left side: \[ 25^3 = 15625 \] - Check the right side: \[ 5^{3+3} = 5^6 = 15625 \] Both sides equal, so \(x = 3\) is the correct solution.

Key Concepts

Base Conversion

Understanding base conversion is an essential aspect of solving exponential equations. At times, you may encounter an equation where both sides have different bases, as was the case in our exercise with bases 25 and 5.
To simplify the problem-solving process, it's often helpful to express both sides using the same base.

• In the given exercise, we observed 25 can be rewritten as a power of 5 since 25 is equivalent to \(5^2\).
• Thus, the left side \(25^x\) can be transformed into \((5^2)^x\), further simplifying to \(5^{2x}\).

Choosing a common base helps us directly compare the exponents of the expressions in the equation. This base conversion simplifies complex problems and makes solving equations more manageable.

Solving Equations

Once the bases in an exponential equation are the same, solving the equation involves equating the exponents.
In our exercise, after converting the expression \(25^x\) to \(5^{2x}\), the equation transformed into \(5^{2x} = 5^{x+3}\).

• Here, since we have the same base on both sides (base 5), we can confidently set the exponents equal: \(2x = x + 3\).
• By doing so, we're left with a straightforward linear equation to solve for \(x\).
• Subtract \(x\) from both sides, simplifying the equation to \(x = 3\).

This approach emphasizes the importance of utilizing base conversion to simplify the problem initially and then focusing on solving the resulting simpler equation.

Exponents

Exponents, which indicate how many times a number (the base) is multiplied by itself, play a fundamental role in exponential equations. Understanding how to manipulate and simplify exponents is crucial.

• The product of powers property, \(a^m \cdot a^n = a^{m+n}\), applies when combining terms with the same base.
• The power of a power property, \((a^m)^n = a^{m\cdot n}\), is useful for simplifying expressions like \((5^2)^x\) into \(5^{2x}\).

Advanced understanding of exponents enables you to efficiently simplify equations like in our problem, ultimately leading to their resolution. Grasping these fundamental properties ensures that you're well-prepared to tackle similar equations.

Verification of Solutions

Once you've solved an exponential equation, verifying your solution is crucial to ensure correctness. This step involves substituting the solution back into the original equation to check if both sides remain equal.

• In our exercise, after finding \(x = 3\), we substitute it back into the original equation \(25^x = 5^{x+3}\).
• Calculating each side, the left becomes \(25^3\) which equals 15625, and the right \(5^{3+3} = 5^6 = 15625\).

Since both sides match, this verification confirms that \(x = 3\) is indeed the correct solution. Verifying your solution provides confidence in your work and is an essential aspect of problem-solving.