Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Solve for \(x\).\(5^{x}=125\)

Step-by-Step Solution

Answer

The solution for the equation is \(x = 3\).

Step 1: Express 125 as a power of 5

First, express 125 as \(5^3\). This allows us to rewrite the original equation \(5^{x} = 125\) as \(5^{x} = 5^3\). Because both sides of the equation have the same base (5), they must also have the same exponent.

Step 2: Set the exponents equal to each other

Next, because both sides of the equation are equal, we can set the exponents equal to each other. This gives us the simplified equation \(x = 3\).

Step 3: Verify the solution

Finally, insert the value of \(x\) back into the initial equation to confirm that it is correct. The equation would be: \(5^{3} = 125\), which is true. Hence, the solution \(x = 3\) is correct.

Key Concepts

Solving Exponential Equations

Exponential equations are expressions where the variable appears in the exponent. To solve these equations, it is crucial to express both sides of the equation with the same base. Let's explore the process with the equation given: \(5^x = 125\). First, the goal is to rewrite the number 125 as a power of 5, so that both sides of the equation have the same base. Since \(125 = 5^3\), the equation becomes \(5^x = 5^3\). Now, with the bases identical, we set the exponents equal to each other: \(x = 3\). This process allows for solving the equation quickly and efficiently. By making the bases the same, we use the property that if \(a^m = a^n\), then \(m = n\).
This understanding turns a seemingly complex equation into a simple linear solution.

Properties of Exponents

Understanding the properties of exponents is essential when working with exponential equations. Here are a few key properties:

• Product of powers: For any non-zero number \(a\), \(a^m \times a^n = a^{m+n}\).
• Quotient of powers: For any non-zero number \(a\), \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a power: For any non-zero number \(a\), \((a^m)^n = a^{mn}\).
• Zero exponent: For any non-zero number \(a\), \(a^0 = 1\).
• Negative exponent: For any non-zero number \(a\), \(a^{-n} = \frac{1}{a^n}\).

Applying these properties helps simplify complex exponential expressions and equations, making it easier to solve problems. Recognizing and utilizing these rules can efficiently transform equations during the problem-solving process.

Checking Solutions in Equations

After solving an exponential equation, it is vital to verify the solution to ensure its accuracy. To check the solution \(x = 3\) for the equation \(5^x = 125\), substitute it back into the original equation. By setting \(5^3\) on the left, we check if it equals 125. Performing the calculation gives us \(5 \times 5 \times 5 = 125\). Since this is correct, it confirms that \(x = 3\) is indeed the right answer.
Verification is a critical step in mathematics, not only for securing the solution but also for gaining confidence in the method used.

• Plug the solution back into the original equation.
• Ensure both sides of the equation balance.
• This step helps prevent errors and solidifies understanding.

Double-checking the solution ensures the integrity of the process and confirms the results confidently.