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

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Introduction

Solving equations with exponents can be challenging, but with the right approach, we can find the solution to the given equation. In this article, we will explore the steps to solve the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ and find the value of $x$.

Understanding the Equation

The given equation is $4\left(\frac{1}{2}\right)^{x-1}=5x+2$. To solve this equation, we need to isolate the variable $x$. The equation involves an exponential term with a base of $\frac{1}{2}$ and a linear term $5x+2$. Our goal is to find the value of $x$ that satisfies the equation.

Step 1: Simplify the Equation

To simplify the equation, we can start by isolating the exponential term. We can do this by dividing both sides of the equation by 4.

$$\left(\frac{1}{2}\right)^{x-1}=\frac{5x+2}{4}$$

Step 2: Take the Logarithm of Both Sides

To eliminate the exponential term, we can take the logarithm of both sides of the equation. We can use the logarithm base 10 or any other base, but we will use the natural logarithm (base $e$) for this example.

$$\ln\left(\left(\frac{1}{2}\right)^{x-1}\right)=\ln\left(\frac{5x+2}{4}\right)$$

Step 3: Apply the Logarithm Property

Using the logarithm property $\ln(a^b)=b\ln(a)$, we can rewrite the equation as:

$$(x-1)\ln\left(\frac{1}{2}\right)=\ln\left(\frac{5x+2}{4}\right)$$

Step 4: Simplify the Equation

We can simplify the equation by dividing both sides by $\ln\left(\frac{1}{2}\right)$.

$$x-1=\frac{\ln\left(\frac{5x+2}{4}\right)}{\ln\left(\frac{1}{2}\right)}$$

Step 5: Solve for x

To solve for $x$, we can add 1 to both sides of the equation.

$$x=\frac{\ln\left(\frac{5x+2}{4}\right)}{\ln\left(\frac{1}{2}\right)}+1$$

Step 6: Use a Numerical Method

Since the equation involves a logarithm and a linear term, we can use a numerical method such as the Newton-Raphson method to find the value of $x$. We can start with an initial guess for $x$ and iteratively update the value until we converge to the solution.

Step 7: Find the Solution

Using a numerical method, we can find the solution to the equation. After several iterations, we get:

$$x\approx\boxed{0.7}$$

Conclusion

In this article, we solved the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ using logarithmic properties and a numerical method. We found that the solution to the equation is $x\approx0.7$.

Discussion

The solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ is $x\approx0.7$. This solution can be verified by plugging it back into the original equation.

Final Answer

The final answer is: $\boxed{0.7}$

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Introduction

In our previous article, we solved the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ using logarithmic properties and a numerical method. In this article, we will answer some frequently asked questions about the solution to the equation.

Q: What is the solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$?

A: The solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ is $x\approx0.7$.

Q: How did you solve the equation?

A: We solved the equation using logarithmic properties and a numerical method. We started by isolating the exponential term, then took the logarithm of both sides of the equation, and finally used a numerical method to find the solution.

Q: What is the significance of the logarithm in solving the equation?

A: The logarithm is used to eliminate the exponential term in the equation. By taking the logarithm of both sides of the equation, we can rewrite the equation in a form that is easier to solve.

Q: Can you explain the numerical method used to solve the equation?

A: The numerical method used to solve the equation is the Newton-Raphson method. This method involves iteratively updating the value of $x$ until we converge to the solution.

Q: How accurate is the solution to the equation?

A: The solution to the equation is accurate to the nearest tenth. This means that the value of $x$ is approximately 0.7, but it may not be exactly 0.7.

Q: Can you provide more information about the Newton-Raphson method?

A: The Newton-Raphson method is a numerical method used to find the roots of a function. It involves iteratively updating the value of the root until we converge to the solution. The method is based on the idea that the root of a function is the point where the function changes sign.

Q: How can I apply the Newton-Raphson method to solve other equations?

A: To apply the Newton-Raphson method to solve other equations, you need to follow these steps:

1. Define the function for which you want to find the root.
2. Choose an initial guess for the root.
3. Iterate the formula for the Newton-Raphson method until you converge to the solution.

Q: What are some common applications of the Newton-Raphson method?

A: The Newton-Raphson method has many applications in science, engineering, and economics. Some common applications include:

• Finding the roots of a polynomial equation
• Solving systems of linear equations
• Finding the maximum or minimum of a function
• Solving optimization problems

Q: Can you provide more information about the logarithmic properties used to solve the equation?

A: The logarithmic properties used to solve the equation are:

• $\ln(a^b)=b\ln(a)$
• $\ln(ab)=\ln(a)+\ln(b)$
• $\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)$

These properties are used to rewrite the equation in a form that is easier to solve.

Q: How can I verify the solution to the equation?

A: To verify the solution to the equation, you can plug it back into the original equation and check if it satisfies the equation.

Conclusion

In this article, we answered some frequently asked questions about the solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$. We provided more information about the logarithmic properties and the numerical method used to solve the equation, and we discussed some common applications of the Newton-Raphson method.

Final Answer

The final answer is: $\boxed{0.7}$