Solve The Equation 64 = 2^(3x) For X.

In the realm of mathematics, exponential equations hold a significant position, often appearing in various scientific and engineering applications. These equations involve variables in the exponents, making them distinct from standard algebraic equations. Mastering the techniques to solve exponential equations is crucial for anyone delving into advanced mathematical concepts. In this comprehensive guide, we will tackle the equation 64 = 2^(3x), providing a step-by-step solution and exploring the underlying principles. This exploration aims to equip you with the knowledge and skills necessary to confidently solve similar problems.

Understanding Exponential Equations

Before diving into the solution, it's essential to grasp the fundamental concept of exponential equations. An exponential equation is one in which the variable appears in the exponent. The general form of an exponential equation is a^x = b, where 'a' is the base, 'x' is the exponent, and 'b' is the result. Solving these equations involves isolating the variable exponent, often by using logarithms or by expressing both sides of the equation with the same base.

In our case, the equation 64 = 2^(3x) presents a classic example of an exponential equation. Our goal is to find the value of 'x' that satisfies this equation. To do this, we will leverage the properties of exponents and logarithms, employing a systematic approach to arrive at the correct solution.

Step-by-Step Solution to 64 = 2^(3x)

Let's break down the solution to the equation 64 = 2^(3x) into a series of clear, manageable steps:

Step 1: Express Both Sides with the Same Base

The key to solving many exponential equations is to express both sides of the equation with the same base. This allows us to equate the exponents directly. In our case, we have 64 on the left side and a power of 2 on the right side. We need to express 64 as a power of 2. We recognize that 64 is 2 raised to the power of 6 (2^6). Therefore, we can rewrite the equation as:

2^6 = 2^(3x)

This step is crucial because it sets the stage for equating the exponents. By expressing both sides with the same base, we transform the problem into a simpler algebraic equation.

Step 2: Equate the Exponents

Now that both sides of the equation have the same base, we can equate the exponents. This is a fundamental property of exponential equations: if a^m = a^n, then m = n. Applying this property to our equation, we get:

6 = 3x

This step significantly simplifies the problem. We have transformed the exponential equation into a linear equation, which is much easier to solve.

Step 3: Solve for x

To solve for 'x', we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by 3:

6 / 3 = (3x) / 3

This simplifies to:

2 = x

Therefore, the solution to the equation 64 = 2^(3x) is x = 2. This completes our step-by-step solution.

Verification of the Solution

It's always a good practice to verify the solution to ensure its correctness. To do this, we substitute x = 2 back into the original equation:

64 = 2^(3 * 2)

64 = 2^6

64 = 64

The equation holds true, confirming that our solution x = 2 is correct. This verification step provides assurance that we have solved the problem accurately.

Alternative Methods: Using Logarithms

While expressing both sides with the same base is a straightforward method for this particular equation, it's not always possible for all exponential equations. In such cases, logarithms provide a powerful alternative. Let's explore how logarithms can be used to solve 64 = 2^(3x).

Step 1: Apply Logarithms to Both Sides

To use logarithms, we take the logarithm of both sides of the equation. The choice of logarithm base is arbitrary, but the common logarithm (base 10) or the natural logarithm (base e) are frequently used. For simplicity, let's use the natural logarithm (ln):

ln(64) = ln(2^(3x))

This step introduces logarithms into the equation, allowing us to leverage their properties.

Step 2: Use the Power Rule of Logarithms

The power rule of logarithms states that ln(a^b) = b * ln(a). Applying this rule to the right side of our equation, we get:

ln(64) = 3x * ln(2)

This step is crucial because it brings the exponent down as a coefficient, making it easier to isolate 'x'.

Step 3: Isolate x

To isolate 'x', we divide both sides of the equation by 3 * ln(2):

ln(64) / (3 * ln(2)) = x

Step 4: Calculate the Value of x

Using a calculator, we can evaluate the logarithmic expressions:

ln(64) ≈ 4.1589

ln(2) ≈ 0.6931

Substituting these values into the equation, we get:

4.1589 / (3 * 0.6931) ≈ x

4.1589 / 2.0793 ≈ x

2 ≈ x

Thus, using logarithms, we arrive at the same solution, x = 2. This demonstrates the versatility of logarithms in solving exponential equations.

Common Mistakes to Avoid

When solving exponential equations, it's essential to be aware of common pitfalls that can lead to incorrect solutions. Here are some mistakes to watch out for:

• Incorrectly Applying the Properties of Exponents: Ensure you correctly apply the rules of exponents, such as the power rule and the product rule. A mistake in applying these rules can lead to a flawed solution.
• Forgetting to Express Both Sides with the Same Base: When using the method of equating exponents, it's crucial to express both sides of the equation with the same base. Failing to do so will prevent you from correctly equating the exponents.
• Misusing Logarithms: When using logarithms, ensure you apply the logarithmic properties correctly. Incorrect application of logarithmic rules, such as the power rule or the quotient rule, can lead to errors.
• Arithmetic Errors: Simple arithmetic errors can derail the solution process. Double-check your calculations to avoid mistakes.
• Not Verifying the Solution: Always verify your solution by substituting it back into the original equation. This step helps catch any errors made during the solving process.

By being mindful of these common mistakes, you can enhance your accuracy and confidence in solving exponential equations.

Real-World Applications of Exponential Equations

Exponential equations are not just theoretical constructs; they have numerous real-world applications across various fields. Here are some examples:

• Population Growth: Exponential equations are used to model population growth, both human and animal. The exponential growth model describes how a population increases over time when the growth rate is proportional to the current population size.
• Compound Interest: The formula for compound interest is an exponential equation. It describes how an investment grows over time when interest is compounded periodically.
• Radioactive Decay: Radioactive decay follows an exponential decay model. The equation describes how the amount of a radioactive substance decreases over time.
• Carbon Dating: Carbon dating, a technique used to determine the age of ancient artifacts, relies on the exponential decay of carbon-14.
• Drug Metabolism: The rate at which a drug is metabolized and eliminated from the body often follows an exponential decay pattern.
• Spread of Diseases: Exponential equations can be used to model the spread of infectious diseases in a population.

These examples illustrate the wide-ranging applicability of exponential equations in understanding and modeling real-world phenomena.

Practice Problems

To solidify your understanding of solving exponential equations, it's essential to practice. Here are a few practice problems for you to try:

1. Solve for x: 8 = 2^(2x)
2. Solve for x: 27 = 3^(x + 1)
3. Solve for x: 125 = 5^(3x - 1)
4. Solve for x: 16 = 4^(x + 2)
5. Solve for x: 100 = 10^(2x + 1)

Try solving these problems using both the method of equating exponents and logarithms. Check your answers to reinforce your learning.

Conclusion

In this comprehensive guide, we have thoroughly explored the process of solving the exponential equation 64 = 2^(3x). We have demonstrated two methods: expressing both sides with the same base and using logarithms. We have also discussed common mistakes to avoid and the real-world applications of exponential equations. By mastering these concepts and techniques, you will be well-equipped to tackle a wide range of exponential equations.

Remember, practice is key to success in mathematics. Work through the practice problems provided and continue to explore more complex equations to enhance your skills. With dedication and perseverance, you can confidently navigate the world of exponential equations and their applications.

The correct answer is B) x=2.