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

Exponential equations are a type of mathematical equation that involves an exponential function. These equations can be challenging to solve, but with the right approach, they can be tackled. In this article, we will explore the solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$ and provide a step-by-step guide on how to solve it.

Understanding Exponential Equations

Exponential equations involve an exponential function, which is a function that raises a number to a power. In the given equation, the exponential function is $\left(\frac{1}{2}\right)^{x-1}$. This function raises the number 12\frac{1}{2} to the power of x−1x-1.

Step 1: Isolate the Exponential Function

To solve the equation, we need to isolate the exponential function. We can do this by dividing both sides of the equation by 4.

4(12)x−14=5x+24\frac{4\left(\frac{1}{2}\right)^{x-1}}{4}=\frac{5x+2}{4}

This simplifies to:

(12)x−1=5x+24\left(\frac{1}{2}\right)^{x-1}=\frac{5x+2}{4}

Step 2: Take the Logarithm of Both Sides

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

ln⁡((12)x−1)=ln⁡(5x+24)\ln\left(\left(\frac{1}{2}\right)^{x-1}\right)=\ln\left(\frac{5x+2}{4}\right)

Using the property of logarithms that states ln⁡(ab)=bln⁡(a)\ln(a^b)=b\ln(a), we can simplify the left-hand side of the equation.

(x−1)ln⁡(12)=ln⁡(5x+24)(x-1)\ln\left(\frac{1}{2}\right)=\ln\left(\frac{5x+2}{4}\right)

Step 3: Simplify the Equation

We can simplify the equation by dividing both sides by ln⁡(12)\ln\left(\frac{1}{2}\right).

x−1=ln⁡(5x+24)ln⁡(12)x-1=\frac{\ln\left(\frac{5x+2}{4}\right)}{\ln\left(\frac{1}{2}\right)}

Step 4: Solve for x

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

x=ln⁡(5x+24)ln⁡(12)+1x=\frac{\ln\left(\frac{5x+2}{4}\right)}{\ln\left(\frac{1}{2}\right)}+1

Step 5: Use a Calculator to Find the Solution

To find the solution to the equation, we can use a calculator to evaluate the expression on the right-hand side.

Using a calculator, we get:

x≈1.6x\approx 1.6

Conclusion

In this article, we have explored the solution to the equation $4\left(\frac{1}{2}\right)^{x-1}=5x+2$. We have used a step-by-step approach to isolate the exponential function, take the logarithm of both sides, simplify the equation, and solve for x. The solution to the equation is x ≈ 1.6.

Answer

The correct answer is C. 1.6.

Discussion

This problem is a great example of how to solve exponential equations. The key is to isolate the exponential function, take the logarithm of both sides, and simplify the equation. With practice, you can become proficient in solving these types of equations.

Additional Resources

If you are struggling with exponential equations, here are some additional resources that may help:

  • Khan Academy: Exponential Equations
  • Mathway: Exponential Equations
  • Wolfram Alpha: Exponential Equations

Final Thoughts

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. Exponential functions are functions that raise a number to a power.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential function, take the logarithm of both sides, and simplify the equation. This can be a challenging process, but with practice, you can become proficient in solving these types of equations.

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

A: An exponential equation involves an exponential function, while a linear equation involves a linear function. Exponential equations can be more challenging to solve than linear equations, but they can also be more interesting and rewarding to solve.

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

A: Yes, you can use a calculator to solve an exponential equation. In fact, calculators can be very helpful in solving these types of equations. However, it's also important to understand the underlying math and to be able to solve the equation by hand.

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 exponential function
  • Not taking the logarithm of both sides
  • Not simplifying the equation correctly
  • Not checking the solution to make sure it is valid

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug the solution back into the original equation and make sure it is true. This can be a good way to verify that your solution is correct.

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

A: Yes, you can use a graphing calculator to solve an exponential equation. Graphing calculators can be very helpful in visualizing the equation and finding the solution.

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

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

  • Modeling population growth
  • Modeling chemical reactions
  • Modeling financial growth
  • Modeling electrical circuits

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

A: Yes, you can use exponential equations to model real-world phenomena. Exponential equations can be used to model a wide range of phenomena, including population growth, chemical reactions, financial growth, and electrical circuits.

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

A: To use exponential equations to model real-world phenomena, you need to:

  • Identify the variables and parameters in the equation
  • Choose an exponential function that fits the data
  • Use the equation to make predictions and forecasts
  • Verify the equation by checking the solution

Q: What are some common types of exponential equations?

A: Some common types of exponential equations include:

  • Exponential growth equations
  • Exponential decay equations
  • Logarithmic equations
  • Power equations

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

A: Yes, you can use exponential equations to solve problems in other areas of mathematics, including:

  • Algebra
  • Geometry
  • Trigonometry
  • Calculus

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

A: To use exponential equations to solve problems in other areas of mathematics, you need to:

  • Identify the variables and parameters in the equation
  • Choose an exponential function that fits the data
  • Use the equation to make predictions and forecasts
  • Verify the equation by checking the solution

Conclusion

Exponential equations are a powerful tool for modeling real-world phenomena and solving problems in mathematics. By understanding how to solve exponential equations, you can gain a deeper understanding of the underlying math and be able to apply it to a wide range of problems.