Type The Correct Answer In The Box. Use Numerals Instead Of Words.What Value Of $x$ Satisfies This Equation? 4 ( 2.5 ) 2 Z = 4 4(2.5)^{2z} = 4 4 ( 2.5 ) 2 Z = 4 The Value Of $x$ Is $\square$.

Mar 13, 2025 by ADMIN 198 views

**Solving Exponential Equations: A Step-by-Step Guide**

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will focus on solving exponential equations of the form $a(b)^{cz} = d$ , where $a$ , $b$ , $c$ , and $d$ are constants. We will use the given equation $4(2.5)^{2z} = 4$ as a case study to illustrate the step-by-step process of solving exponential equations.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and the goal is to isolate the variable. In the given equation, $4(2.5)^{2z} = 4$ , we have a base of $2.5$ raised to the power of $2z$ . The constant $4$ is multiplied by the exponential term, resulting in an equation that requires careful manipulation to isolate the variable $z$ .

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by dividing both sides by $4$ . This will eliminate the constant term and leave us with an exponential equation.

\frac{4(2.5)^{2z}}{4} = \frac{4}{4}

\Rightarrow (2.5)^{2z} = 1

Step 2: Use Logarithms to Isolate the Variable

To isolate the variable $z$ , we can use logarithms to bring the exponent down. We will use the natural logarithm (ln) to solve for $z$ .

\ln((2.5)^{2z}) = \ln(1)

\Rightarrow 2z \ln(2.5) = 0

Step 3: Solve for the Variable

Now that we have isolated the variable $z$ , we can solve for its value.

2z \ln(2.5) = 0

\Rightarrow z = \frac{0}{2 \ln(2.5)}

\Rightarrow z = 0

Conclusion

In this article, we have demonstrated the step-by-step process of solving exponential equations. We used the given equation $4(2.5)^{2z} = 4$ as a case study to illustrate the process. By simplifying the equation, using logarithms to isolate the variable, and solving for the variable, we arrived at the solution $z = 0$ . This example highlights the importance of careful manipulation and the use of logarithms in solving exponential equations.

Final Answer

The value of $x$ that satisfies the equation is $\boxed{0}$ .

Discussion

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we have demonstrated the step-by-step process of solving exponential equations. If you have any questions or would like to discuss further, please feel free to leave a comment below.

Additional Resources

Mathematics Textbook: A comprehensive textbook covering various topics in mathematics, including exponential equations.
Online Math Resources: A collection of online resources, including videos, tutorials, and practice problems, to help you improve your math skills.
Mathematics Community: A community forum where you can ask questions, share knowledge, and learn from others in the mathematics community.

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Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves a variable raised to a power. It is typically written in the form $a(b)^{cz} = d$ , where $a$ , $b$ , $c$ , and $d$ are constants.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable by using algebraic manipulations and logarithms. The steps involved in solving an exponential equation are:

Simplify the equation by dividing both sides by the constant term.
Use logarithms to bring the exponent down.
Solve for the variable.

Q: What is the difference between an exponential equation and a linear equation?

A: An exponential equation involves a variable raised to a power, whereas a linear equation involves a variable multiplied by a constant. For example, $2x = 4$ is a linear equation, while $2^x = 4$ is an exponential equation.

Q: Can I use logarithms to solve any exponential equation?

A: Yes, you can use logarithms to solve any exponential equation. However, you need to choose the correct type of logarithm (natural logarithm or common logarithm) and apply the correct properties of logarithms to solve the equation.

Q: What is the significance of the base in an exponential equation?

A: The base in an exponential equation is the number that is raised to a power. The base determines the rate at which the variable changes. For example, in the equation $2^x = 4$ , the base is 2, which means that the variable $x$ changes at a rate of 2.

Q: Can I use algebraic manipulations to solve an exponential equation?

A: Yes, you can use algebraic manipulations to solve an exponential equation. However, you need to be careful when using algebraic manipulations, as they can lead to incorrect solutions.

Q: What is the difference between an exponential function and an exponential equation?

A: An exponential function is a mathematical function that involves a variable raised to a power, whereas an exponential equation is a mathematical equation that involves a variable raised to a power. For example, $f(x) = 2^x$ is an exponential function, while $2^x = 4$ is an exponential equation.

Q: Can I use technology to solve an exponential equation?

A: Yes, you can use technology, such as calculators or computer software, to solve an exponential equation. However, you need to understand the underlying mathematics and use the technology correctly to arrive at the correct solution.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

Not simplifying the equation correctly
Not using the correct type of logarithm
Not applying the correct properties of logarithms
Not checking the solution for extraneous solutions

Q: How do I check my solution for extraneous solutions?

A: To check your solution for extraneous solutions, you need to plug the solution back into the original equation and verify that it is true. If the solution is not true, then it is an extraneous solution and should be discarded.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

Modeling population growth
Modeling financial growth
Modeling chemical reactions
Modeling electrical circuits

Q: Can I use exponential equations to model real-world phenomena?

A: Yes, you can use exponential equations to model real-world phenomena. However, you need to understand the underlying mathematics and use the exponential equation correctly to arrive at a realistic model.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

Equations with a base of 2
Equations with a base of 10
Equations with a base of e (the natural base)
Equations with a variable base

Q: Can I use exponential equations to solve problems in other areas of mathematics?

A: Yes, you can use exponential equations to solve problems in other areas of mathematics, including:

Algebra
Geometry
Trigonometry
Calculus

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

Simplifying the equation correctly
Using the correct type of logarithm
Applying the correct properties of logarithms
Checking the solution for extraneous solutions

Q: Can I use exponential equations to solve problems in science and engineering?

A: Yes, you can use exponential equations to solve problems in science and engineering, including:

Modeling population growth
Modeling financial growth
Modeling chemical reactions
Modeling electrical circuits

Q: What are some common mistakes to avoid when using exponential equations in science and engineering?

A: Some common mistakes to avoid when using exponential equations in science and engineering include:

Not simplifying the equation correctly
Not using the correct type of logarithm
Not applying the correct properties of logarithms
Not checking the solution for extraneous solutions

Q: How do I choose the correct type of logarithm to use in an exponential equation?

A: To choose the correct type of logarithm to use in an exponential equation, you need to consider the base of the exponential equation and the type of logarithm that is most convenient to use. For example, if the base is 2, you may want to use the common logarithm (log), while if the base is e, you may want to use the natural logarithm (ln).