Problem 11 - Advanced Engineering Mathematics

Question

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Step-by-Step Solution

Answer

The value of \(x\) is 3.

Step 1: Understanding the Problem

We are given the problem to solve: \(5^x = 125\). The goal is to find the value of \(x\) that makes this equation true.

Step 2: Recognizing the Exponential Relationship

Notice that on the right side of the equation, 125 can be expressed as a power of 5. Since \(5^3 = 125\), we can rewrite the equation as \(5^x = 5^3\).

Step 3: Equating the Exponents

Since the bases are the same (both are 5), we can set the exponents equal to each other. Therefore, \(x = 3\).

Step 4: Solution Verification

Substitute \(x = 3\) back into the original equation to verify: \(5^3 = 125\). This equality holds true, confirming that \(x = 3\) is the correct solution.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation where a number, known as the base, is raised to an exponent or power, symbolizing repeated multiplication of that number. For example, in the expression \(5^x\), 5 is the base raised to the power of \(x\). This means we multiply 5 by itself \(x\) times. Exponentiation is crucial for simplifying expressions and solving equations efficiently. It's used to represent very large or very small numbers in a compact form.
Here’s a quick list to understand the terms in exponentiation:

• **Base**: The number that gets multiplied (e.g., 5 in \(5^x\)).
• **Exponent**: The number that shows how many times to multiply the base (e.g., \(x\) in the expression \(5^x\)).

Exponentiation extends beyond whole number exponents to include fractions and negatives, which change the meaning of the operation. In solving exponential equations, recognizing these relationships helps us rewrite expressions and equations in more manageable forms.

Solving Equations

In mathematics, solving equations involves finding the value of the unknown variable that makes the equation true. This exercise was an example of solving exponential equations where both sides of the equation had the same base, making it easier to solve.
Consider the equation \(5^x = 125\). Here, solving the equation involves the following steps:

• **Express in terms of the same base**: Recognize that 125 is a power of 5, specifically \(5^3\).
• **Equate the exponents**: Since the bases are identical, set the exponents equal to each other so \(x = 3\).

This method ensures that we isolate the variable effectively. Such strategies are central to solving exponential equations and can be applied to various mathematical problems where exponential terms are involved.

Mathematical Verification

Mathematical verification is the process of confirming the correctness of a found solution. It’s an important part of solving any equation because it ensures that our calculations are accurate and the solution is valid.
In verifying a solution:

• **Substitute the solution back into the original equation**: For instance, substituting \(x = 3\) into \(5^x = 125\) gives us \(5^3 = 125\), which is true.
• **Double-check calculations**: Ensure every step aligns with mathematical principles.
• **Observe consistency**: The outcomes of substituting back should be logical and consistent with the original conditions of the problem.

Verification not only helps in confirming our answers but also builds confidence in the application of mathematical concepts.

Mathematics Education

Mathematics education aims to develop students’ skills in analysis, critical thinking, and problem-solving through various mathematical concepts, including exponential equations. Understanding these basic principles lays the foundation for more complex subjects.
Important aspects of teaching and learning mathematics include:

• **Conceptual understanding**: Grasping how and why procedures and concepts work.
• **Procedural fluency**: Developing skills for carrying out procedures flexibly and accurately.
• **Application**: Using mathematical concepts in various contexts to solve real-world problems.

By linking these core ideas with exercises, students can see the relevance and utility of mathematics in everyday life, fostering a deeper interest and competence in the subject.