Matching Each Equation to the Correct Answer

Oct 13, 2025 by leolavanderia.com.br 86 views

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will explore four exponential equations and match each equation to the correct answer. We will use the properties of exponents, such as the product rule and the power rule, to solve each equation.

Equation 1: $5^{x-3} = 25^{z-5}$

To solve this equation, we can start by rewriting both sides with the same base. Since $25 = 5^2$ , we can rewrite the equation as:

5^{x-3} = (5^2)^{z-5}

Using the power rule, we can rewrite the right-hand side as:

5^{x-3} = 5^{2(z-5)}

Since the bases are the same, we can equate the exponents:

x-3 = 2(z-5)

Simplifying the equation, we get:

x-3 = 2z-10

Adding 3 to both sides, we get:

x = 2z-7

Equation 2: $e^{2x} = e^{3x-1}$

To solve this equation, we can start by using the property of equality of exponential functions with the same base. Since the bases are the same, we can equate the exponents:

2x = 3x-1

Subtracting $3x$ from both sides, we get:

-x = -1

Dividing both sides by $-1$ , we get:

x = 1

Equation 3: $6^{2x-6} = 36^{3x-5}$

To solve this equation, we can start by rewriting both sides with the same base. Since $36 = 6^2$ , we can rewrite the equation as:

6^{2x-6} = (6^2)^{3x-5}

Using the power rule, we can rewrite the right-hand side as:

6^{2x-6} = 6^{2(3x-5)}

Simplifying the exponent, we get:

6^{2x-6} = 6^{6x-10}

Since the bases are the same, we can equate the exponents:

2x-6 = 6x-10

Adding 6 to both sides, we get:

2x = 6x-4

Subtracting $6x$ from both sides, we get:

-4x = -4

Dividing both sides by $-4$ , we get:

x = 1

Equation 4: $7^{3x+5} = 7^{1-x}$

To solve this equation, we can start by using the property of equality of exponential functions with the same base. Since the bases are the same, we can equate the exponents:

3x+5 = 1-x

Adding $x$ to both sides, we get:

4x+5 = 1

Subtracting 5 from both sides, we get:

4x = -4

Dividing both sides by 4, we get:

x = -1

Matching Each Equation to the Correct Answer

Now that we have solved each equation, we can match each equation to the correct answer.

Equation 1: $5^{x-3} = 25^{z-5}$ matches with answer b. $x = 2z-7$
Equation 2: $e^{2x} = e^{3x-1}$ matches with answer a. $x = 1$
Equation 3: $6^{2x-6} = 36^{3x-5}$ matches with answer a. $x = 1$
Equation 4: $7^{3x+5} = 7^{1-x}$ matches with answer a. $x = -1$

Conclusion

In this article, we have solved four exponential equations and matched each equation to the correct answer. We have used the properties of exponents, such as the product rule and the power rule, to solve each equation. We have also used the property of equality of exponential functions with the same base to equate the exponents. By following these steps, we have been able to solve each equation and match each equation to the correct answer.

Final Answer

The final answer is:

Equation 1: b. $x = 2z-7$
Equation 2: a. $x = 1$
Equation 3: a. $x = 1$
Equation 4: a. $x = -1$

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will answer some frequently asked questions about exponential equations and provide examples to illustrate the concepts.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, $2^x$ is an exponential expression, where $2$ is the base and $x$ is the exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable (usually $x$ ) by using the properties of exponents. There are several properties of exponents that you can use to solve exponential equations, including:

The product rule: $a^m \cdot a^n = a^{m+n}$
The power rule: $(a^m)^n = a^{mn}$
The quotient rule: $\frac{a^m}{a^n} = a^{m-n}$
The equality of exponential functions with the same base: If $a^x = a^y$ , then $x = y$

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

A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression. A logarithmic expression is the inverse of an exponential expression, and it is used to solve exponential equations.

Q: How do I use the properties of exponents to solve an exponential equation?

A: To use the properties of exponents to solve an exponential equation, you need to apply the properties in the correct order. Here are the steps:

Simplify the exponential expression by using the product rule, power rule, or quotient rule.
Use the equality of exponential functions with the same base to equate the exponents.
Solve for the variable (usually $x$ ) by isolating it on one side of the equation.

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

A: Here are some common mistakes to avoid when solving exponential equations:

Not simplifying the exponential expression before solving for the variable.
Not using the correct properties of exponents to solve the equation.
Not isolating the variable on one side of the equation.
Not checking the solution to make sure it is valid.

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug the solution back into the original equation and make sure it is true. If the solution is not true, then you need to re-solve the equation.

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

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

Modeling population growth and decline.
Modeling the spread of diseases.
Modeling the growth of investments.
Modeling the decay of radioactive materials.

Q: How do I use technology to solve exponential equations?

A: There are many software programs and online tools that can help you solve exponential equations, including:

Graphing calculators.
Online equation solvers.
Spreadsheets.
Programming languages.

Conclusion

In this article, we have answered some frequently asked questions about exponential equations and provided examples to illustrate the concepts. We have also discussed the properties of exponents and how to use them to solve exponential equations. By following the steps outlined in this article, you should be able to solve exponential equations with ease.

Final Answer

The final answer is:

Q: What is an exponential equation? A: An exponential equation is an equation that involves an exponential expression.
Q: How do I solve an exponential equation? A: To solve an exponential equation, you need to isolate the variable (usually $x$ ) by using the properties of exponents.
Q: What is the difference between an exponential equation and a logarithmic equation? A: An exponential equation is an equation that involves an exponential expression, while a logarithmic equation is an equation that involves a logarithmic expression.
Q: How do I use the properties of exponents to solve an exponential equation? A: To use the properties of exponents to solve an exponential equation, you need to apply the properties in the correct order.
Q: What are some common mistakes to avoid when solving exponential equations? A: Here are some common mistakes to avoid when solving exponential equations: not simplifying the exponential expression, not using the correct properties of exponents, not isolating the variable, and not checking the solution.
Q: How do I check my solution to an exponential equation? A: To check your solution to an exponential equation, you need to plug the solution back into the original equation and make sure it is true.
Q: What are some real-world applications of exponential equations? A: Exponential equations have many real-world applications, including modeling population growth and decline, modeling the spread of diseases, modeling the growth of investments, and modeling the decay of radioactive materials.
Q: How do I use technology to solve exponential equations? A: There are many software programs and online tools that can help you solve exponential equations, including graphing calculators, online equation solvers, spreadsheets, and programming languages.