Which Of The Following Could Be The First Step In Solving The Equation Below? 5 X = 21 5^x=21 5 X = 21 A. Log ⁡ 5 X = 21 \log 5^x=21 Lo G 5 X = 21 B. Log ⁡ 5 X = Log ⁡ 21 \log 5^x=\log 21 Lo G 5 X = Lo G 21 C. 5 X 5 = 21 5 \sqrt[5]{5^x}=\sqrt[5]{21} 5 5 X ​ = 5 21 ​ D. 1 5 ⋅ 5 X = 1 5 ⋅ 21 \frac{1}{5} \cdot 5^x=\frac{1}{5} \cdot 21 5 1 ​ ⋅ 5 X = 5 1 ​ ⋅ 21

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithmic and exponential functions. In this article, we will explore the first step in solving the equation 5x=215^x=21. We will examine four possible options and determine which one is the correct first step.

Understanding Exponential Equations

Exponential equations involve a variable raised to a power, and the equation is set equal to a constant or another expression. In the case of the equation 5x=215^x=21, we are trying to find the value of xx that makes the equation true. To solve this equation, we need to use logarithmic properties and manipulate the equation to isolate the variable.

Option A: log5x=21\log 5^x=21

This option involves taking the logarithm of both sides of the equation. However, this is not the correct first step in solving the equation. Taking the logarithm of both sides would result in log5x=log21\log 5^x = \log 21, which is not a useful step in solving the equation.

Option B: log5x=log21\log 5^x=\log 21

This option also involves taking the logarithm of both sides of the equation. However, this is not the correct first step in solving the equation. Taking the logarithm of both sides would result in log5x=log21\log 5^x = \log 21, which is not a useful step in solving the equation.

Option C: 5x5=215\sqrt[5]{5^x}=\sqrt[5]{21}

This option involves taking the fifth root of both sides of the equation. This is a useful step in solving the equation, as it allows us to eliminate the exponent and isolate the variable. By taking the fifth root of both sides, we get 5x5=215\sqrt[5]{5^x} = \sqrt[5]{21}, which is a more manageable equation.

Option D: 155x=1521\frac{1}{5} \cdot 5^x=\frac{1}{5} \cdot 21

This option involves multiplying both sides of the equation by 15\frac{1}{5}. However, this is not the correct first step in solving the equation. Multiplying both sides by 15\frac{1}{5} would result in a more complicated equation, rather than a simpler one.

Conclusion

In conclusion, the correct first step in solving the equation 5x=215^x=21 is to take the fifth root of both sides of the equation. This results in 5x5=215\sqrt[5]{5^x} = \sqrt[5]{21}, which is a more manageable equation. By taking the fifth root of both sides, we can eliminate the exponent and isolate the variable, making it easier to solve the equation.

Why This Step is Important

Taking the fifth root of both sides of the equation is an important step in solving the equation 5x=215^x=21. This step allows us to eliminate the exponent and isolate the variable, making it easier to solve the equation. By taking the fifth root of both sides, we can use logarithmic properties to solve the equation and find the value of xx.

Real-World Applications

Exponential equations have many real-world applications, including finance, science, and engineering. In finance, exponential equations are used to model population growth and compound interest. In science, exponential equations are used to model chemical reactions and population growth. In engineering, exponential equations are used to model the behavior of electrical circuits and mechanical systems.

Common Mistakes to Avoid

When solving exponential equations, there are several common mistakes to avoid. One mistake is to take the logarithm of both sides of the equation without considering the base of the logarithm. Another mistake is to multiply both sides of the equation by a constant without considering the effect on the equation. By avoiding these common mistakes, we can ensure that we are taking the correct first step in solving the equation.

Final Thoughts

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable raised to a power, and the equation is set equal to a constant or another expression. For example, the equation 5x=215^x=21 is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to use logarithmic properties and manipulate the equation to isolate the variable. One common method is to take the logarithm of both sides of the equation, but this is not always the best approach.

Q: What is the correct first step in solving the equation 5x=215^x=21?

A: The correct first step in solving the equation 5x=215^x=21 is to take the fifth root of both sides of the equation. This results in 5x5=215\sqrt[5]{5^x} = \sqrt[5]{21}, which is a more manageable equation.

Q: Why is taking the fifth root of both sides of the equation a good approach?

A: Taking the fifth root of both sides of the equation is a good approach because it allows us to eliminate the exponent and isolate the variable. This makes it easier to solve the equation and find the value of xx.

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

A: Some common mistakes to avoid when solving exponential equations include taking the logarithm of both sides of the equation without considering the base of the logarithm, and multiplying both sides of the equation by a constant without considering the effect on the equation.

Q: How do I use logarithmic properties to solve an exponential equation?

A: To use logarithmic properties to solve an exponential equation, you need to take the logarithm of both sides of the equation and use the properties of logarithms to simplify the equation. For example, if you have the equation 5x=215^x=21, you can take the logarithm of both sides and use the property logab=bloga\log a^b = b \log a to simplify the equation.

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

A: Exponential equations have many real-world applications, including finance, science, and engineering. In finance, exponential equations are used to model population growth and compound interest. In science, exponential equations are used to model chemical reactions and population growth. In engineering, exponential equations are used to model the behavior of electrical circuits and mechanical systems.

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

A: To check your answer when solving an exponential equation, you need to plug your solution back into the original equation and verify that it is true. For example, if you have the equation 5x=215^x=21 and you think the solution is x=2x=2, you can plug x=2x=2 back into the equation and verify that 52=215^2=21 is true.

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

  • Always read the problem carefully and understand what is being asked.
  • Use logarithmic properties to simplify the equation and isolate the variable.
  • Check your answer by plugging it back into the original equation.
  • Avoid common mistakes such as taking the logarithm of both sides of the equation without considering the base of the logarithm, and multiplying both sides of the equation by a constant without considering the effect on the equation.

Q: How do I practice solving exponential equations?

A: To practice solving exponential equations, you can try solving a variety of problems using different bases and exponents. You can also try using online resources such as calculators and worksheets to help you practice. Additionally, you can try solving problems that involve real-world applications of exponential equations, such as modeling population growth or compound interest.