Consider The Equation \[$-2 \cdot 10^{4x} = -300\$\].1. Solve The Equation For \[$x\$\]. Express The Solution As A Logarithm In Base-10. $\[ X = \square \\]2. Approximate The Value Of \[$x\$\]. Round Your Answer To

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions. In this article, we will explore how to solve the equation βˆ’2β‹…104x=βˆ’300{-2 \cdot 10^{4x} = -300} for x{x}, and express the solution as a logarithm in base-10.

Step 1: Isolate the Exponential Term

The first step in solving the equation is to isolate the exponential term. We can do this by dividing both sides of the equation by -2.

βˆ’2β‹…104x=βˆ’300{-2 \cdot 10^{4x} = -300}

Dividing both sides by -2 gives us:

104x=βˆ’300βˆ’2104x=150{\begin{aligned} 10^{4x} &= \frac{-300}{-2} \\ 10^{4x} &= 150 \end{aligned}}

Step 2: Use Logarithms to Solve for x

Now that we have isolated the exponential term, we can use logarithms to solve for x. We can take the logarithm base-10 of both sides of the equation to get:

log⁑10(104x)=log⁑10(150)4x=log⁑10(150){\begin{aligned} \log_{10} (10^{4x}) &= \log_{10} (150) \\ 4x &= \log_{10} (150) \end{aligned}}

Step 3: Simplify the Equation

Now that we have simplified the equation, we can solve for x by dividing both sides by 4.

4x=log⁑10(150)x=log⁑10(150)4{\begin{aligned} 4x &= \log_{10} (150) \\ x &= \frac{\log_{10} (150)}{4} \end{aligned}}

Step 4: Approximate the Value of x

Now that we have the solution in terms of a logarithm, we can approximate the value of x by plugging in the value of log⁑10(150){\log_{10} (150)} into the equation.

x=log⁑10(150)4x=2.1764x=0.544{\begin{aligned} x &= \frac{\log_{10} (150)}{4} \\ x &= \frac{2.176}{4} \\ x &= 0.544 \end{aligned}}

Conclusion

In this article, we have shown how to solve the equation βˆ’2β‹…104x=βˆ’300{-2 \cdot 10^{4x} = -300} for x{x}, and express the solution as a logarithm in base-10. We have also approximated the value of x by plugging in the value of log⁑10(150){\log_{10} (150)} into the equation. The final answer is x=0.544{x = 0.544}.

Discussion

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions. In this article, we have shown how to solve the equation βˆ’2β‹…104x=βˆ’300{-2 \cdot 10^{4x} = -300} for x{x}, and express the solution as a logarithm in base-10. We have also approximated the value of x by plugging in the value of log⁑10(150){\log_{10} (150)} into the equation.

Solving Exponential Equations: Tips and Tricks

  • Make sure to isolate the exponential term before using logarithms to solve for x.
  • Use the change of base formula to convert between different logarithmic bases.
  • Approximate the value of x by plugging in the value of the logarithm into the equation.

Common Mistakes to Avoid

  • Failing to isolate the exponential term before using logarithms to solve for x.
  • Using the wrong logarithmic base.
  • Approximating the value of x without plugging in the value of the logarithm into the equation.

Real-World Applications

Exponential equations have many real-world applications, including:

  • Modeling population growth and decay.
  • Calculating compound interest.
  • Solving problems involving exponential decay.

Conclusion

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic functions. In this article, we will explore some common questions and answers 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 number raised to a power. For example, 2x=8{2^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 use logarithms to solve for the variable. For example, to solve the equation 2x=8{2^x = 8}, you would first isolate the exponential term by dividing both sides by 2, and then use logarithms to solve for x.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to convert between different logarithmic bases. It is given by:

log⁑b(x)=log⁑c(x)log⁑c(b){\log_b (x) = \frac{\log_c (x)}{\log_c (b)}}

where b and c are the bases of the logarithms.

Q: How do I use the change of base formula?

A: To use the change of base formula, you need to identify the bases of the logarithms in the equation and then plug them into the formula. For example, to solve the equation log⁑2(x)=3{\log_2 (x) = 3}, you would first convert the logarithm to base 10 using the change of base formula, and then solve for x.

Q: What is the difference between a logarithmic equation 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, log⁑2(x)=3{\log_2 (x) = 3} is a logarithmic equation. An exponential equation, on the other hand, is an equation that involves an exponential expression. For example, 2x=8{2^x = 8} is an exponential equation.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term and then use the properties of logarithms to solve for the variable. For example, to solve the equation log⁑2(x)=3{\log_2 (x) = 3}, you would first isolate the logarithmic term by raising both sides to the power of 2, and then solve for x.

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

A: Some common mistakes to avoid when solving exponential equations include:

  • Failing to isolate the exponential term before using logarithms to solve for x.
  • Using the wrong logarithmic base.
  • Approximating the value of x without plugging in the value of the logarithm into the equation.

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

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

  • Modeling population growth and decay.
  • Calculating compound interest.
  • Solving problems involving exponential decay.

Conclusion

In conclusion, solving exponential equations requires a deep understanding of logarithmic functions. By following the steps outlined in this article, you can solve exponential equations and apply them to real-world problems. Remember to avoid common mistakes and use the change of base formula to convert between different logarithmic bases.

Additional Resources

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

Practice Problems

  1. Solve the equation 2x=16{2^x = 16} for x.
  2. Solve the equation log⁑2(x)=4{\log_2 (x) = 4} for x.
  3. Solve the equation 3x=27{3^x = 27} for x.

Answers

  1. x = 4
  2. x = 16
  3. x = 3