Solving Exponential Equations (1/5000)^(-2z) * 5000^(-2z+2) = 5000 A Comprehensive Guide

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Introduction

In this article, we will delve into the step-by-step solution of the exponential equation ${\left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000}$. Exponential equations, which involve variables in the exponents, often appear complex, but they can be systematically solved using the properties of exponents and algebraic manipulation. We will explore each step in detail, providing a clear and comprehensive understanding of the solution process. This guide aims to equip you with the skills necessary to tackle similar problems confidently.

Problem Statement

We are given the equation:

${ \left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000 }$

Our objective is to find the value(s) of ${ z }$ that satisfy this equation. We will break down the solution into manageable steps, ensuring each transformation is clearly justified and explained.

Step-by-Step Solution

Step 1: Rewrite the Fraction as a Negative Exponent

The first step involves recognizing that ${\frac{1}{5,000}}$ can be expressed as ${5,000^{-1}}$. This transformation allows us to consolidate the terms with the same base, which is crucial for simplifying the equation. By rewriting the fraction, we make it easier to apply the properties of exponents.

${ \left(5,000^{-1}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000 }$

Step 2: Apply the Power of a Power Rule

The power of a power rule states that ${(a^m)^n = a^{mn}}$. Applying this rule to the first term, ${\left(5,000^{-1}\right)^{-2 z}}$, we multiply the exponents ${-1}$ and ${-2z}$, resulting in ${5,000^{2z}}$. This simplification is a key step in combining like terms and moving towards isolating the variable.

${ 5,000^{2 z} \cdot 5,000^{-2 z+2}=5,000 }$

Step 3: Apply the Product of Powers Rule

The product of powers rule states that ${a^m \cdot a^n = a^{m+n}}$. Now, we apply this rule to the left side of the equation. We add the exponents ${2z}$ and ${-2z + 2}$ of the terms ${5,000^{2z}}$ and ${5,000^{-2z+2}}$. This step consolidates the terms with the same base into a single term, making the equation simpler to solve.

${ 5,000^{2 z + (-2 z + 2)}=5,000 }$

Step 4: Simplify the Exponent

Next, we simplify the exponent by combining like terms. The exponent on the left side of the equation is ${2z + (-2z + 2)}$. The ${2z}$ and ${-2z}$ terms cancel each other out, leaving us with just the constant 2. This simplification significantly reduces the complexity of the equation.

${ 5,000^{2}=5,000 }$

Step 5: Analyze the Simplified Equation

At this point, we have the simplified equation ${5,000^{2}=5,000}$. This equation states that 5,000 raised to the power of 2 is equal to 5,000. However, this is not true since ${5,000^2 = 25,000,000}$, which is clearly not equal to 5,000. This contradiction indicates that there is no solution for ${ z }$ that satisfies the original equation.

Conclusion

Therefore, based on our step-by-step simplification and analysis, we conclude that there is no solution for the equation ${\left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000}$. This corresponds to option D in the given choices.

Final Answer

The final answer is (D) no solution.

Detailed Explanation of Exponential Equations

To further understand the problem and its solution, it is essential to grasp the fundamentals of exponential equations. Exponential equations are mathematical equations in which variables occur in the exponents. These equations often require a strong understanding of the properties of exponents to solve effectively. In this section, we will delve deeper into the properties and techniques used to solve such equations.

Properties of Exponents

The properties of exponents are the cornerstone of solving exponential equations. Understanding and applying these properties correctly is crucial for simplifying and solving these equations. Here are some fundamental properties that are frequently used:

1. Product of Powers Rule: When multiplying two exponential expressions with the same base, you add the exponents. Mathematically, this is represented as:

   ${ a^m \cdot a^n = a^{m+n} }$

   Example: ${2^3 \cdot 2^4 = 2^{3+4} = 2^7}$

2. Quotient of Powers Rule: When dividing two exponential expressions with the same base, you subtract the exponents:

   ${ \frac{a^m}{a^n} = a^{m-n} }$

   Example: ${\frac{3^5}{3^2} = 3^{5-2} = 3^3}$

3. Power of a Power Rule: When raising an exponential expression to another power, you multiply the exponents:

   ${ (a^m)^n = a^{mn} }$

   Example: ${(4^2)^3 = 4^{2 \cdot 3} = 4^6}$

4. Power of a Product Rule: When raising a product to a power, you raise each factor to the power:

   ${ (ab)^n = a^n b^n }$

   Example: ${(2 \cdot 5)^3 = 2^3 \cdot 5^3}$

5. Power of a Quotient Rule: When raising a quotient to a power, you raise both the numerator and the denominator to the power:

   ${ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} }$

   Example: ${\left(\frac{3}{4}\right)^2 = \frac{3^2}{4^2}}$

6. Negative Exponent Rule: A negative exponent indicates a reciprocal:

   ${ a^{-n} = \frac{1}{a^n} }$

   Example: ${5^{-2} = \frac{1}{5^2} = \frac{1}{25}}$

7. Zero Exponent Rule: Any non-zero number raised to the power of 0 is 1:

   ${ a^0 = 1 \quad \text{for } a \neq 0 }$

   Example: ${7^0 = 1}$

Techniques for Solving Exponential Equations

1. Expressing Both Sides with the Same Base

One of the most common techniques for solving exponential equations is to express both sides of the equation with the same base. Once the bases are the same, you can equate the exponents and solve for the variable. This technique is particularly effective when the numbers involved can be easily expressed as powers of the same base.

Example: Solve ${2^{3x} = 8}$

Solution: First, express 8 as a power of 2, which is ${2^3}$. The equation becomes ${2^{3x} = 2^3}$. Now, equate the exponents: ${3x = 3}$. Solving for ${x}$, we get ${x = 1}$.

2. Using Logarithms

When it is not straightforward to express both sides with the same base, logarithms can be used. Logarithms are the inverse operation to exponentiation and provide a powerful tool for solving exponential equations. The most common logarithms used are the natural logarithm (base ${e}$) and the common logarithm (base 10).

Example: Solve ${5^x = 15}$

Solution: Take the natural logarithm (ln) of both sides: (\ln(5^x) = \ln(15)). Using the power rule of logarithms, ${x \ln(5) = \ln(15)}$. Now, solve for ${x}$: ${x = \frac{\ln(15)}{\ln(5)}}$. This can be further approximated using a calculator.

3. Substitution

In some cases, exponential equations can be simplified using substitution. This involves replacing a complex exponential expression with a single variable, solving for the variable, and then substituting back to find the original variable. This technique is particularly useful when dealing with equations that have repeated exponential terms.

Example: Solve ${4^x - 6 \cdot 2^x + 8 = 0}$

Solution: Notice that ${4^x}$ can be written as ${(2^2)^x = (2^x)^2}$. Let ${y = 2^x}$. The equation becomes ${y^2 - 6y + 8 = 0}$. This is a quadratic equation that can be factored as ${(y - 4)(y - 2) = 0}$. Thus, ${y = 4}$ or ${y = 2}$. Now substitute back: If ${2^x = 4}$, then ${x = 2}$. If ${2^x = 2}$, then ${x = 1}$.

4. Using Properties of Equality

The properties of equality, such as the addition, subtraction, multiplication, and division properties, can be used to manipulate exponential equations. These properties allow you to perform the same operation on both sides of the equation, maintaining the equality and moving towards a solution.

Common Mistakes to Avoid

1. Incorrectly Applying Exponent Rules: A common mistake is misapplying the properties of exponents. For example, students may incorrectly add exponents when they should be multiplying them, or vice versa. Always double-check the exponent rules before applying them.

2. Forgetting to Distribute Exponents: When raising a product or quotient to a power, remember to distribute the exponent to each factor or term. For example, ${(ab)^n = a^n b^n}$, not ${ab^n}$.

3. Not Checking for Extraneous Solutions: When using logarithms, it is essential to check for extraneous solutions. Logarithms are only defined for positive arguments, so any solution that results in taking the logarithm of a negative number or zero is extraneous and must be discarded.

4. Misunderstanding the Base: Always ensure that you are working with the correct base when simplifying exponential expressions. For example, ${2^{3x}}$ is different from ${3^{2x}}$, and they cannot be combined directly.

Advanced Techniques and Considerations

1. Complex Exponential Equations

Some exponential equations may involve more complex expressions or require advanced techniques, such as calculus or numerical methods, to solve. These types of equations are often encountered in advanced mathematics and engineering applications.

2. Exponential Inequalities

In addition to equations, exponential expressions can also appear in inequalities. Solving exponential inequalities involves similar techniques as solving equations, but with the added consideration of the direction of the inequality. For example, when the base is greater than 1, the inequality sign remains the same when equating exponents, but when the base is between 0 and 1, the inequality sign is reversed.

3. Applications of Exponential Equations

Exponential equations have numerous applications in various fields, including finance, biology, physics, and computer science. They are used to model phenomena such as population growth, radioactive decay, compound interest, and algorithm complexity. Understanding how to solve exponential equations is essential for analyzing and solving real-world problems in these areas.

Practice Problems

To reinforce your understanding of solving exponential equations, here are some practice problems with varying levels of difficulty:

1. Solve: ${3^{2x} = 81}$
2. Solve: ${2^{x+1} = 32}$
3. Solve: ${5^{x-2} = 125}$
4. Solve: ${4^x = 16^{x-1}}$
5. Solve: ${9^x = 3^{x+1}}$
6. Solve: ${2^{2x} - 5 \cdot 2^x + 4 = 0}$
7. Solve: ${3^{2x} - 10 \cdot 3^x + 9 = 0}$
8. Solve: ${5^{2x} - 6 \cdot 5^x + 5 = 0}$
9. Solve: ${2^x = 10}$ (Use logarithms)
10. Solve: ${3^x = 20}$ (Use logarithms)

Solutions to Practice Problems

1. ${x = 2}$
2. ${x = 4}$
3. ${x = 5}$
4. ${x = 2}$
5. ${x = 1}$
6. ${x = 0, 2}$
7. ${x = 0, 2}$
8. ${x = 0, 1}$
9. ${x = \frac{\ln(10)}{\ln(2)} \approx 3.3219}$
10. ${x = \frac{\ln(20)}{\ln(3)} \approx 2.7268}$

By working through these practice problems and their solutions, you can solidify your understanding of the techniques for solving exponential equations and improve your problem-solving skills. Remember to carefully apply the properties of exponents and logarithms, and always check your answers to ensure they are correct.

Conclusion

In summary, solving exponential equations requires a solid understanding of the properties of exponents and various techniques such as expressing both sides with the same base, using logarithms, and substitution. The problem ${\left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000}$ illustrates a scenario where simplification leads to a contradiction, indicating no solution. By mastering these concepts and practicing regularly, you can confidently tackle a wide range of exponential equations and related problems.