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

In this article, we will explore the solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$. This equation involves an exponential term and a linear term, making it a challenging problem to solve. We will use algebraic techniques to isolate the variable and find the solution.

Understanding the Equation

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

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 Exponent Properties

Next, we can use exponent properties to simplify the equation further. We can rewrite the exponential term as $\left(\frac{1}{2}\right)^{x-1}=\left(\frac{1}{2}\right)^{x}\left(\frac{1}{2}\right)^{-1}$.

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

Step 3: Simplify the Exponential Term

We can simplify the exponential term by using the property $\left(\frac{1}{2}\right)^{-1}=2$.

$$8\left(\frac{1}{2}\right)^{x}\cdot2=10x+4$$

Step 4: Simplify the Equation

Now, we can simplify the equation by combining like terms.

$$16\left(\frac{1}{2}\right)^{x}=10x+4$$

Step 5: Take the Logarithm

To solve for $x$, we can take the logarithm of both sides of the equation. We will use the natural logarithm (ln) for this purpose.

$$\ln\left(16\left(\frac{1}{2}\right)^{x}\right)=\ln\left(10x+4\right)$$

Step 6: Use Logarithm Properties

Next, we can use logarithm properties to simplify the equation further. We can rewrite the logarithm of a product as the sum of the logarithms.

$$\ln\left(16\right)+\ln\left(\left(\frac{1}{2}\right)^{x}\right)=\ln\left(10x+4\right)$$

Step 7: Simplify the Logarithm

We can simplify the logarithm by using the property $\ln\left(a^{b}\right)=b\ln\left(a\right)$.

$$\ln\left(16\right)+x\ln\left(\frac{1}{2}\right)=\ln\left(10x+4\right)$$

Step 8: Simplify the Equation

Now, we can simplify the equation by combining like terms.

$$4.0-x\ln\left(\frac{1}{2}\right)=\ln\left(10x+4\right)$$

Step 9: Isolate the Variable

To isolate the variable $x$, we can move all the terms involving $x$ to one side of the equation.

$$x\ln\left(\frac{1}{2}\right)=4.0-\ln\left(10x+4\right)$$

Step 10: Solve for x

Finally, we can solve for $x$ by dividing both sides of the equation by $\ln\left(\frac{1}{2}\right)$.

$$x=\frac{4.0-\ln\left(10x+4\right)}{\ln\left(\frac{1}{2}\right)}$$

Numerical Solution

To find the numerical solution, we can use a numerical method such as the Newton-Raphson method. This method involves making an initial guess for the solution and then iteratively improving the guess until the solution converges.

Using the Newton-Raphson method, we find that the solution to the equation is $x\approx\boxed{0.7}$.

Conclusion

In this article, we have solved the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ using algebraic techniques. We have used exponent properties, logarithm properties, and numerical methods to isolate the variable and find the solution. The solution to the equation is $x\approx\boxed{0.7}$.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Numerical Methods for Scientists and Engineers" by Richard Hamming

Discussion

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Introduction

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

Q: What is the solution to the equation?

A: The solution to the equation is $x\approx\boxed{0.7}$.

Q: How did you solve the equation?

A: We used algebraic techniques, including exponent properties, logarithm properties, and numerical methods to isolate the variable and find the solution.

Q: What is the significance of the logarithm in the solution?

A: The logarithm is used to simplify the equation and isolate the variable. It helps to eliminate the exponential term and make the equation more manageable.

Q: Can you explain the Newton-Raphson method used to find the numerical solution?

A: The Newton-Raphson method is a numerical method used to find the solution to an equation. It involves making an initial guess for the solution and then iteratively improving the guess until the solution converges. In this case, we used the Newton-Raphson method to find the numerical solution to the equation.

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

A: Some common mistakes to avoid when solving equations like this include:

• Not simplifying the equation enough before trying to solve it
• Not using the correct properties of logarithms and exponents
• Not checking the solution to make sure it is valid
• Not using numerical methods when necessary

Q: Can you provide more information about the properties of logarithms and exponents used in the solution?

A: The properties of logarithms and exponents used in the solution include:

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

Q: Can you explain the concept of convergence in the context of the Newton-Raphson method?

A: Convergence in the context of the Newton-Raphson method refers to the process of iteratively improving the guess for the solution until it converges to the actual solution. In other words, the solution is said to converge when the guess is close enough to the actual solution that further iterations do not change the solution.

Conclusion

In this article, we have answered some frequently asked questions about the solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$. We have discussed the significance of the logarithm in the solution, the Newton-Raphson method used to find the numerical solution, and some common mistakes to avoid when solving equations like this. We hope this article has been helpful in clarifying any questions or concerns you may have had about the solution to the equation.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Numerical Methods for Scientists and Engineers" by Richard Hamming

Discussion

Do you have any questions or comments about the solution to the equation or the Q&A article? Please share your thoughts in the discussion section below.