Find The Solution To The Exponential Equation:$ \frac{17}{2}(2)^x = 272 }$Choose The Correct Solution A. The Solution Is { X = 11$ $.B. The Solution Is { X = 5$}$.C. The Solution Is { X = 8$}$.D. The Solution

Mar 11, 2025 by ADMIN 210 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 algebraic manipulations and properties of exponents. In this article, we will focus on solving the exponential equation $\frac{17}{2}(2)^x = 272$ and provide a step-by-step guide to finding the correct solution.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and they can be written in the form $a^x = b$ , where $a$ and $b$ are constants, and $x$ is the variable. In the given equation, $\frac{17}{2}(2)^x = 272$ , we have a base of $2$ raised to the power of $x$ , multiplied by a constant $\frac{17}{2}$ .

Isolating the Variable

To solve the equation, we need to isolate the variable $x$ . We can start by dividing both sides of the equation by $\frac{17}{2}$ to get rid of the constant multiplier.

\frac{\frac{17}{2}(2)^x}{\frac{17}{2}} = \frac{272}{\frac{17}{2}}

This simplifies to:

(2)^x = \frac{272}{\frac{17}{2}}

Simplifying the Right-Hand Side

To simplify the right-hand side, we can multiply the numerator and denominator by $2$ to get rid of the fraction.

\frac{272}{\frac{17}{2}} = \frac{272 \times 2}{17}

This simplifies to:

(2)^x = \frac{544}{17}

Using Logarithms to Solve for x

Now that we have isolated the variable $x$ and simplified the right-hand side, we can use logarithms to solve for $x$ . We can take the logarithm of both sides of the equation to get:

\log(2)^x = \log\left(\frac{544}{17}\right)

Using the property of logarithms that $\log(a^b) = b \log(a)$ , we can rewrite the equation as:

x \log(2) = \log\left(\frac{544}{17}\right)

Solving for x

To solve for $x$ , we can divide both sides of the equation by $\log(2)$ .

x = \frac{\log\left(\frac{544}{17}\right)}{\log(2)}

Evaluating the Expression

To evaluate the expression, we can use a calculator to find the values of the logarithms.

x = \frac{\log\left(\frac{544}{17}\right)}{\log(2)} \approx \frac{2.732}{0.301} \approx 9.07

Choosing the Correct Solution

Now that we have solved for $x$ , we can compare our solution to the options provided.

A. The solution is $x = 11$ .
B. The solution is $x = 5$ .
C. The solution is $x = 8$ .
D. The solution is $x = 9.07$ .

Based on our calculation, the correct solution is:

D. The solution is $x = 9.07$

Conclusion

Solving exponential equations requires a deep understanding of algebraic manipulations and properties of exponents. By following the step-by-step guide provided in this article, we were able to solve the equation $\frac{17}{2}(2)^x = 272$ and find the correct solution. Remember to always use logarithms to solve for the variable $x$ when dealing with exponential equations.

Final Answer

The final answer is: $\boxed{9.07}$

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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, and it can be written in the form $a^x = b$ , where $a$ and $b$ are constants, and $x$ is the variable.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable $x$ by using algebraic manipulations and properties of exponents. You can start by dividing both sides of the equation by a constant multiplier, and then use logarithms to solve for $x$ .

Q: What is the property of logarithms that I can use to solve for x?

A: The property of logarithms that you can use to solve for $x$ is $\log(a^b) = b \log(a)$ . This allows you to rewrite the equation in a form that is easier to solve.

Q: How do I use a calculator to evaluate the expression for x?

A: To evaluate the expression for $x$ , you can use a calculator to find the values of the logarithms. For example, if you have the equation $x = \frac{\log\left(\frac{544}{17}\right)}{\log(2)}$ , you can use a calculator to find the values of the logarithms and then divide to find the value of $x$ .

Q: What is the correct solution to the equation $\frac{17}{2}(2)^x = 272$ ?

A: The correct solution to the equation $\frac{17}{2}(2)^x = 272$ is $x = 9.07$ .

Q: Can I use other methods to solve exponential equations?

A: Yes, there are other methods that you can use to solve exponential equations, such as using the change of base formula or using a graphing calculator. However, the method of using logarithms is often the most straightforward and easiest to use.

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

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

Not isolating the variable $x$ correctly
Not using the correct property of logarithms
Not evaluating the expression for $x$ correctly
Not checking the solution to make sure it is correct

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving real-world problems that involve exponential equations, such as calculating the growth or decay of a population or the value of an investment.

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

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

Calculating the growth or decay of a population
Calculating the value of an investment
Modeling the spread of a disease
Calculating the amount of money in a savings account or a loan

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

A: Yes, exponential equations can be used to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry. For example, you can use exponential equations to solve problems involving exponential growth or decay, or to model real-world phenomena that involve exponential relationships.

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

Equations with a base of $e$ (such as $e^x = 5$ )
Equations with a base of $a$ (such as $2^x = 8$ )
Equations with a base of $b$ (such as $3^x = 27$ )
Equations with a base of $c$ (such as $4^x = 64$ )

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

A: You can use exponential equations to model real-world phenomena by identifying the variables and constants in the equation and using them to represent the real-world situation. For example, you can use an exponential equation to model the growth or decay of a population, or to calculate the value of an investment.

Q: What are some common mistakes to avoid when using exponential equations to model real-world phenomena?

A: Some common mistakes to avoid when using exponential equations to model real-world phenomena include:

Not identifying the variables and constants correctly
Not using the correct property of logarithms
Not evaluating the expression for $x$ correctly
Not checking the solution to make sure it is correct

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

A: You can use exponential equations to solve problems in science and engineering by identifying the variables and constants in the equation and using them to represent the real-world situation. For example, you can use an exponential equation to model the growth or decay of a population, or to calculate the value of an investment.

Q: What are some common types of exponential equations used in science and engineering?

A: Some common types of exponential equations used in science and engineering include:

Equations with a base of $e$ (such as $e^x = 5$ )
Equations with a base of $a$ (such as $2^x = 8$ )
Equations with a base of $b$ (such as $3^x = 27$ )
Equations with a base of $c$ (such as $4^x = 64$ )

Q: How can I use exponential equations to solve problems in finance?

A: You can use exponential equations to solve problems in finance by identifying the variables and constants in the equation and using them to represent the real-world situation. For example, you can use an exponential equation to calculate the value of an investment or to model the growth or decay of a portfolio.

Q: What are some common types of exponential equations used in finance?

A: Some common types of exponential equations used in finance include:

Equations with a base of $e$ (such as $e^x = 5$ )
Equations with a base of $a$ (such as $2^x = 8$ )
Equations with a base of $b$ (such as $3^x = 27$ )
Equations with a base of $c$ (such as $4^x = 64$ )

Q: How can I use exponential equations to solve problems in economics?

A: You can use exponential equations to solve problems in economics by identifying the variables and constants in the equation and using them to represent the real-world situation. For example, you can use an exponential equation to model the growth or decay of a population, or to calculate the value of an investment.

Q: What are some common types of exponential equations used in economics?

A: Some common types of exponential equations used in economics include:

Equations with a base of $e$ (such as $e^x = 5$ )
Equations with a base of $a$ (such as $2^x = 8$ )
Equations with a base of $b$ (such as $3^x = 27$ )
Equations with a base of $c$ (such as $4^x = 64$ )

Q: How can I use exponential equations to solve problems in computer science?

A: You can use exponential equations to solve problems in computer science by identifying the variables and constants in the equation and using them to represent the real-world situation. For example, you can use an exponential equation to model the growth or decay of a population, or to calculate the value of an investment.

Q: What are some common types of exponential equations used in computer science?

A: Some common types of exponential equations used in computer science include:

Equations with a base of $e$ (such as $e^x = 5$ )
Equations with a base of $a$ (such as $2^x = 8$ )
Equations with a base of $b$ (such as $3^x = 27$ )
Equations with a base of $c$ (such as $4^x = 64$ )

Q: How can I use exponential equations to solve problems in data analysis?

A: You can use exponential equations to solve problems in data analysis by identifying the variables and constants in the equation and using them to represent the real-world situation. For example, you can use an exponential equation to model the growth or decay of a population, or to calculate the value of an investment.

Q: What are some common types of exponential equations used in data analysis?

A: Some common types of exponential equations used in data analysis include:

Equations with a base of $e$ (such as $e^x = 5$ )
Equations with a base of $a$ (such as $2^x = 8$ )
Equations with a base of $b$ (such as $3^x = 27$ )
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