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 and solve for x.

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

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 Solution

Now, we can use a calculator to find the solution to the equation. 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 solution. In this case, we used algebraic techniques to isolate the variable and solve for x. We also used a calculator to find the solution and rounded the answer to the nearest tenth.

Introduction

Solving equations with exponents can be a challenging task, but with the right techniques and strategies, you can find the solution. In this article, we will answer some frequently asked questions about solving equations with exponents.

Q: What is the first step in solving an equation with an exponent?

A: The first step in solving an equation with an exponent 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 deal with a negative exponent in an equation?

A: When dealing with a negative exponent in an equation, you can rewrite the equation by taking the reciprocal of both sides. For example, if you have the equation 2^(-x) = 3, you can rewrite it as 2^x = 1/3.

Q: Can I use logarithms to solve an equation with an exponent?

A: Yes, you can use logarithms to solve an equation with an exponent. By taking the logarithm of both sides of the equation, you can get rid of the exponent and solve for the variable.

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

A: A linear equation is an equation in which the variable is raised to the power of 1, while an exponential equation is an equation in which the variable is raised to a power other than 1.

Q: How do I solve an equation with a base of 10?

A: To solve an equation with a base of 10, you can use the fact that 10^x = 10^(x-1) * 10. This can help you simplify the equation and solve for the variable.

Q: Can I use a calculator to solve an equation with an exponent?

A: Yes, you can use a calculator to solve an equation with an exponent. However, you should always check your answer to make sure it is correct.

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

A: The most common mistake people make when solving equations with exponents is not simplifying the equation enough before solving for the variable.

Q: How do I check my answer when solving an equation with an exponent?

A: To check your answer when solving an equation with an exponent, you can plug the solution back into the original equation and make sure it is true.

Q: Can I use a graphing calculator to solve an equation with an exponent?

A: Yes, you can use a graphing calculator to solve an equation with an exponent. However, you should always check your answer to make sure it is correct.

Q: What is the best way to learn how to solve equations with exponents?

A: The best way to learn how to solve equations with exponents is to practice, practice, practice. Start with simple equations and work your way up to more complex ones.

Conclusion

Solving equations with exponents can be a challenging task, but with the right techniques and strategies, you can find the solution. By following the steps outlined in this article and practicing regularly, you can become proficient in solving equations with exponents.

Additional Resources

• Mathway: A free online math problem solver that can help you solve equations with exponents.
• Khan Academy: A free online learning platform that offers video lessons and practice exercises on solving equations with exponents.
• Wolfram Alpha: A free online calculator that can help you solve equations with exponents.

Final Thoughts

Solving equations with exponents is an important skill that can be applied to a wide range of real-world problems. By mastering this skill, you can become a more confident and proficient math student. Remember to practice regularly and seek help when you need it. With time and effort, you can become a math whiz!