Solve The Equation:${ \begin{array}{c} 20 = 100\left(\frac{1}{2}\right)^{\frac{x}{214}} \ x = , ? \end{array} }$Round To The Nearest Hundredth.

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic and exponential functions. In this article, we will focus on solving a specific exponential equation, which involves a fraction raised to a power. We will break down the solution into manageable steps, making it easier for readers to understand and apply the concepts.

The Equation

The given equation is:

20=100(12)x21420 = 100\left(\frac{1}{2}\right)^{\frac{x}{214}}

Our goal is to solve for xx and round the answer to the nearest hundredth.

Step 1: Isolate the Exponential Term

To solve the equation, we need to isolate the exponential term. We can start by dividing both sides of the equation by 100:

20100=(12)x214\frac{20}{100} = \left(\frac{1}{2}\right)^{\frac{x}{214}}

This simplifies to:

15=(12)x214\frac{1}{5} = \left(\frac{1}{2}\right)^{\frac{x}{214}}

Step 2: Take the Logarithm of Both Sides

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

ln⁡(15)=ln⁡((12)x214)\ln\left(\frac{1}{5}\right) = \ln\left(\left(\frac{1}{2}\right)^{\frac{x}{214}}\right)

Using the property of logarithms that states ln⁡(ab)=bln⁡(a)\ln(a^b) = b\ln(a), we can rewrite the right-hand side of the equation as:

ln⁡(15)=x214ln⁡(12)\ln\left(\frac{1}{5}\right) = \frac{x}{214}\ln\left(\frac{1}{2}\right)

Step 3: Simplify the Logarithmic Terms

We can simplify the logarithmic terms by using the fact that ln⁡(1/a)=−ln⁡(a)\ln(1/a) = -\ln(a):

ln⁡(15)=−ln⁡(5)\ln\left(\frac{1}{5}\right) = -\ln(5)

ln⁡(12)=−ln⁡(2)\ln\left(\frac{1}{2}\right) = -\ln(2)

Substituting these values into the equation, we get:

−ln⁡(5)=x214(−ln⁡(2))-\ln(5) = \frac{x}{214}(-\ln(2))

Step 4: Solve for x

To solve for xx, we can multiply both sides of the equation by −214-214:

214ln⁡(5)=xln⁡(2)214\ln(5) = x\ln(2)

Now, we can divide both sides of the equation by ln⁡(2)\ln(2) to isolate xx:

x=214ln⁡(5)ln⁡(2)x = \frac{214\ln(5)}{\ln(2)}

Step 5: Calculate the Value of x

Using a calculator, we can calculate the value of xx:

x≈214×1.6094379120.6931471806x \approx \frac{214 \times 1.609437912}{0.6931471806}

x≈344.11111110.6931471806x \approx \frac{344.1111111}{0.6931471806}

x≈495.0000000x \approx 495.0000000

Conclusion

In this article, we solved an exponential equation involving a fraction raised to a power. We broke down the solution into manageable steps, making it easier for readers to understand and apply the concepts. The final answer is x≈495.00x \approx 495.00, rounded to the nearest hundredth.

Additional Tips and Resources

  • When solving exponential equations, it's essential to isolate the exponential term and then take the logarithm of both sides.
  • Use the properties of logarithms to simplify the equation and make it easier to solve.
  • A calculator can be a valuable tool when solving exponential equations, especially when dealing with complex calculations.
  • For more information on exponential equations and logarithms, check out the following resources:
  • Khan Academy: Exponential and Logarithmic Functions
  • Mathway: Exponential Equations
  • Wolfram Alpha: Exponential Equations

Final Answer

Introduction

In our previous article, we solved an exponential equation involving a fraction raised to a power. We broke down the solution into manageable steps, making it easier for readers to understand and apply the concepts. In this article, we will answer some frequently asked questions (FAQs) related to solving exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a power of a base raised to a variable exponent. For example, the equation 2x=82^x = 8 is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the exponential term and then take the logarithm of both sides. This will allow you to eliminate the exponent and solve for the variable.

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

A: A logarithmic equation is an equation that involves a logarithmic expression, which is the inverse of an exponential expression. For example, the equation log⁡2(x)=3\log_2(x) = 3 is a logarithmic equation. An exponential equation, on the other hand, involves an exponential expression.

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations. In fact, a calculator can be a valuable tool when dealing with complex calculations. However, it's essential to understand the underlying concepts and principles to ensure that you are using the calculator correctly.

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 term
  • Not taking the logarithm of both sides
  • Not using the correct base for the logarithm
  • Not checking the domain of the logarithmic function

Q: Can I use different types of logarithms to solve exponential equations?

A: Yes, you can use different types of logarithms to solve exponential equations. For example, you can use the natural logarithm (ln), the common logarithm (log), or the base-10 logarithm (log10). The choice of logarithm depends on the specific problem and the base of the exponential expression.

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

A: To check your answer when solving an exponential equation, you can plug the solution back into the original equation and verify that it is true. You can also use a calculator to check the solution.

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

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

  • Population growth and decay
  • Financial calculations, such as compound interest
  • Physics and engineering problems, such as radioactive decay and exponential growth
  • Computer science and programming, such as exponential time complexity

Conclusion

In this article, we answered some frequently asked questions related to solving exponential equations. We covered topics such as the definition of an exponential equation, how to solve an exponential equation, and common mistakes to avoid. We also discussed the use of different types of logarithms and how to check your answer when solving an exponential equation. We hope that this article has been helpful in clarifying any questions you may have had about solving exponential equations.

Additional Resources

  • Khan Academy: Exponential and Logarithmic Functions
  • Mathway: Exponential Equations
  • Wolfram Alpha: Exponential Equations
  • MIT OpenCourseWare: Exponential Functions and Equations

Final Answer

The final answer is: There is no final numerical answer to this article, as it is a Q&A guide.