What Is The Solution To The Equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$? Round To The Nearest Tenth.A. 0.6 B. 0.7 C. 1.6 D. 5.2

Introduction

Solving equations involving exponents can be challenging, especially when the exponent is part of the equation. In this case, we have an equation with a base of 1/2 and an exponent of x-1. Our goal is to find the value of x that satisfies the equation. We will use algebraic techniques to isolate the variable x and find its value.

Step 1: Simplify the Equation

The first step is to simplify the equation by getting rid of the fraction. We can do this by multiplying both sides of the equation by 2.

$$8\left(\frac{1}{2}\right)^{x-1}=10x+4$$

Step 2: Use the Properties of Exponents

Next, we can use the properties of exponents to simplify the left-hand side of the equation. Specifically, we can use the fact that (1/2)^(x-1) = (1/2)^x / (1/2).

$$8\left(\frac{1}{2}\right)^x / (1/2)=10x+4$$

Step 3: Simplify the Left-Hand Side

Now, we can simplify the left-hand side of the equation by multiplying both sides by (1/2).

$$4\left(\frac{1}{2}\right)^x=10x+4$$

Step 4: Take the Logarithm of Both Sides

To get rid of the exponent, we can take the logarithm of both sides of the equation. We will use the natural logarithm (ln) for this purpose.

$$\ln(4\left(\frac{1}{2}\right)^x)=\ln(10x+4)$$

Step 5: Use the Properties of Logarithms

Next, we can use the properties of logarithms to simplify the left-hand side of the equation. Specifically, we can use the fact that ln(a^b) = b*ln(a).

$$x\ln(1/2)+\ln(4)=\ln(10x+4)$$

Step 6: Isolate the Variable x

Now, we can isolate the variable x by subtracting ln(4) from both sides of the equation and then dividing both sides by ln(1/2).

$$x=\frac{\ln(10x+4)-\ln(4)}{\ln(1/2)}$$

Step 7: Simplify the Expression

To simplify the expression, we can use the fact that ln(a) - ln(b) = ln(a/b).

$$x=\frac{\ln\left(\frac{10x+4}{4}\right)}{\ln(1/2)}$$

Step 8: Use a Calculator to Find the Value of x

Now, we can use a calculator to find the value of x. We will round the answer to the nearest tenth.

$$x=\frac{\ln\left(\frac{10x+4}{4}\right)}{\ln(1/2)}\approx\boxed{0.7}$$

The final answer is 0.7.

Conclusion

Solving equations involving exponents can be challenging, but with the right techniques, we can find the value of the variable. In this case, we used algebraic techniques to isolate the variable x and then used a calculator to find its value. The final answer is 0.7.

Discussion

This problem is a good example of how to solve equations involving exponents. The key is to use the properties of exponents and logarithms to simplify the equation and isolate the variable. With practice, you can become proficient in solving these types of equations.

Related Problems

If you want to practice solving equations involving exponents, try the following problems:

• Solve the equation $2^x=3x+2$
• Solve the equation $5^x=2x+1$
• Solve the equation $3^x=x2+1$

These problems are similar to the one we solved in this article, but with different bases and exponents. With practice, you can become proficient in solving these types of equations.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

These references provide a comprehensive introduction to algebra and calculus, including equations involving exponents. They are a good starting point for anyone who wants to learn more about this topic.

Introduction

Solving equations involving exponents can be challenging, but with the right techniques and practice, you can become proficient in solving these types of equations. In this article, we will answer some common questions about solving equations involving exponents.

Q: What is the first step in solving an equation involving exponents?

A: The first step in solving an equation involving exponents is to simplify the equation by getting rid of any fractions or decimals. This can be done by multiplying both sides of the equation by a common factor.

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

A: To use the properties of exponents to simplify an equation, you need to identify the base and the exponent. Then, you can use the fact that (a^b)c = a^(bc) to simplify the equation.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, the equation 2^x = 10 is an exponential equation, while the equation log(2) = x is a logarithmic equation.

Q: How do I solve an equation that involves a logarithm?

A: To solve an equation that involves a logarithm, you need to use the properties of logarithms to simplify the equation. Specifically, you can use the fact that log(a) + log(b) = log(ab) and log(a) - log(b) = log(a/b).

Q: What is the most common mistake people make when solving equations involving exponents?

A: The most common mistake people make when solving equations involving exponents is to forget to check their work. It's easy to get caught up in the math and forget to plug in the solution to the original equation to make sure it's true.

Q: How can I practice solving equations involving exponents?

A: There are many ways to practice solving equations involving exponents. You can start by working through practice problems in a textbook or online resource. You can also try solving equations involving exponents on your own, using a calculator to check your work.

Q: What are some common types of equations involving exponents?

A: Some common types of equations involving exponents include:

• Exponential equations: equations that involve an exponent, such as 2^x = 10
• Logarithmic equations: equations that involve a logarithm, such as log(2) = x
• Equations with multiple bases: equations that involve multiple bases, such as 2^x = 3^x
• Equations with multiple exponents: equations that involve multiple exponents, such as 2^x = 3^(2x)

Q: How can I use technology to help me solve equations involving exponents?

A: There are many ways to use technology to help you solve equations involving exponents. You can use a calculator to check your work, or use a computer program to solve the equation for you. You can also use online resources, such as math websites or apps, to practice solving equations involving exponents.

Conclusion

Solving equations involving exponents can be challenging, but with the right techniques and practice, you can become proficient in solving these types of equations. By following the steps outlined in this article, you can learn how to solve equations involving exponents and become a more confident math student.

Discussion

If you have any questions or need further clarification on any of the topics discussed in this article, please don't hesitate to ask. We would be happy to help you understand the concepts and provide additional practice problems to help you improve your skills.

Related Problems

If you want to practice solving equations involving exponents, try the following problems:

• Solve the equation $2^x=3x+2$
• Solve the equation $5^x=2x+1$
• Solve the equation $3^x=x2+1$

These problems are similar to the ones we solved in this article, but with different bases and exponents. With practice, you can become proficient in solving these types of equations.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

These references provide a comprehensive introduction to algebra and calculus, including equations involving exponents. They are a good starting point for anyone who wants to learn more about this topic.