What Is The True Solution To The Logarithmic Equation? Log ⁡ 2 [ Log ⁡ 2 ( 4 X ) ] = 1 \log_2\left[\log_2(\sqrt{4x})\right] = 1 Lo G 2 ​ [ Lo G 2 ​ ( 4 X ​ ) ] = 1 A. X = − 4 X = -4 X = − 4 B. X = 0 X = 0 X = 0 C. X = 2 X = 2 X = 2 D. X = 4 X = 4 X = 4

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Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will explore the solution to the logarithmic equation log⁡2[log⁡2(4x)]=1\log_2\left[\log_2(\sqrt{4x})\right] = 1. This equation involves multiple layers of logarithms, making it a challenging problem to solve. We will break down the solution step by step, using the properties of logarithms to simplify the equation and find the value of xx.

Understanding the Equation

The given equation is log⁡2[log⁡2(4x)]=1\log_2\left[\log_2(\sqrt{4x})\right] = 1. To solve this equation, we need to understand the properties of logarithms. The logarithm of a number is the exponent to which a base must be raised to produce that number. In this case, the base is 2, and the logarithm is log⁡2\log_2. The equation involves multiple layers of logarithms, making it a complex problem to solve.

Properties of Logarithms

Before we dive into the solution, let's review some important properties of logarithms:

  • Logarithm of a product: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
  • Logarithm of a quotient: log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)
  • Logarithm of a power: log⁡b(xy)=ylog⁡b(x)\log_b(x^y) = y \log_b(x)
  • Change of base formula: log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}

These properties will be essential in simplifying the equation and finding the value of xx.

Step 1: Simplify the Inner Logarithm

The inner logarithm is log⁡2(4x)\log_2(\sqrt{4x}). We can simplify this using the property of logarithms: log⁡b(x)=12log⁡b(x)\log_b(\sqrt{x}) = \frac{1}{2} \log_b(x). Applying this property, we get:

log⁡2(4x)=12log⁡2(4x)\log_2(\sqrt{4x}) = \frac{1}{2} \log_2(4x)

Step 2: Simplify the Outer Logarithm

The outer logarithm is log⁡2[12log⁡2(4x)]=1\log_2\left[\frac{1}{2} \log_2(4x)\right] = 1. We can simplify this using the property of logarithms: log⁡b(12x)=log⁡b(x)−log⁡b(2)\log_b(\frac{1}{2}x) = \log_b(x) - \log_b(2). Applying this property, we get:

log⁡2[12log⁡2(4x)]=log⁡2(log⁡2(4x))−log⁡2(2)\log_2\left[\frac{1}{2} \log_2(4x)\right] = \log_2(\log_2(4x)) - \log_2(2)

Step 3: Equate the Expression to 1

Since the equation is log⁡2[log⁡2(4x)]=1\log_2\left[\log_2(\sqrt{4x})\right] = 1, we can equate the expression to 1:

log⁡2(log⁡2(4x))−log⁡2(2)=1\log_2(\log_2(4x)) - \log_2(2) = 1

Step 4: Simplify the Equation

We can simplify the equation by adding log⁡2(2)\log_2(2) to both sides:

log⁡2(log⁡2(4x))=1+log⁡2(2)\log_2(\log_2(4x)) = 1 + \log_2(2)

Step 5: Exponentiate Both Sides

We can exponentiate both sides using the base 2:

log⁡2(4x)=21⋅2log⁡2(2)\log_2(4x) = 2^1 \cdot 2^{\log_2(2)}

Step 6: Simplify the Right-Hand Side

We can simplify the right-hand side using the property of exponents: alog⁡a(b)=ba^{\log_a(b)} = b. Applying this property, we get:

log⁡2(4x)=2⋅2\log_2(4x) = 2 \cdot 2

Step 7: Simplify the Equation

We can simplify the equation by multiplying both sides:

log⁡2(4x)=4\log_2(4x) = 4

Step 8: Exponentiate Both Sides

We can exponentiate both sides using the base 2:

4x=244x = 2^4

Step 9: Simplify the Right-Hand Side

We can simplify the right-hand side using the property of exponents: ab=c  ⟹  b=log⁡a(c)a^b = c \implies b = \log_a(c). Applying this property, we get:

4x=164x = 16

Step 10: Solve for x

We can solve for xx by dividing both sides by 4:

x=164x = \frac{16}{4}

Step 11: Simplify the Right-Hand Side

We can simplify the right-hand side by dividing both numbers:

x=4x = 4

The final answer is 4\boxed{4}.

Conclusion

In this article, we explored the solution to the logarithmic equation log⁡2[log⁡2(4x)]=1\log_2\left[\log_2(\sqrt{4x})\right] = 1. We broke down the solution step by step, using the properties of logarithms to simplify the equation and find the value of xx. The final answer is x=4x = 4. This solution demonstrates the importance of understanding the properties of logarithms and how to apply them to solve complex equations.

Frequently Asked Questions

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm is 2.

Q: What is the property of logarithms used to simplify the inner logarithm?

A: The property of logarithms used to simplify the inner logarithm is log⁡b(x)=12log⁡b(x)\log_b(\sqrt{x}) = \frac{1}{2} \log_b(x).

Q: What is the property of logarithms used to simplify the outer logarithm?

A: The property of logarithms used to simplify the outer logarithm is log⁡b(12x)=log⁡b(x)−log⁡b(2)\log_b(\frac{1}{2}x) = \log_b(x) - \log_b(2).

Q: What is the final answer to the equation?

A: The final answer to the equation is x=4x = 4.

References

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Introduction

Logarithmic equations can be challenging to solve, but understanding the properties of logarithms and how to apply them can make a big difference. In this article, we will answer some frequently asked questions about logarithmic equations, providing a deeper understanding of the subject.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. Logarithmic equations can be used to solve problems involving growth and decay, as well as to model real-world phenomena.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

  • Logarithm of a product: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)
  • Logarithm of a quotient: log⁡b(xy)=log⁡b(x)−log⁡b(y)\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)
  • Logarithm of a power: log⁡b(xy)=ylog⁡b(x)\log_b(x^y) = y \log_b(x)
  • Change of base formula: log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms to combine the terms. For example, if you have log⁡b(x)+log⁡b(y)\log_b(x) + \log_b(y), you can simplify it to log⁡b(xy)\log_b(xy).

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to isolate the variable. For example, if you have log⁡b(x)=y\log_b(x) = y, you can exponentiate both sides to get x=byx = b^y.

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

A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent. For example, log⁡b(x)=y\log_b(x) = y is a logarithmic equation, while x=byx = b^y is an exponential equation.

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

A: Yes, logarithmic equations can be used to model real-world phenomena, such as population growth, chemical reactions, and financial transactions.

Q: What are some common applications of logarithmic equations?

A: Some common applications of logarithmic equations include:

  • Population growth: Logarithmic equations can be used to model population growth and decline.
  • Chemical reactions: Logarithmic equations can be used to model chemical reactions and predict the outcome of a reaction.
  • Financial transactions: Logarithmic equations can be used to model financial transactions and predict the outcome of a investment.

Q: How do I choose the base of a logarithm?

A: The base of a logarithm is usually chosen to be a positive number greater than 1. The most common bases are 2, 10, and e.

Q: What is the difference between a common logarithm and a natural logarithm?

A: A common logarithm is a logarithm with a base of 10, while a natural logarithm is a logarithm with a base of e.

Q: Can I use logarithmic equations to solve problems involving fractions?

A: Yes, logarithmic equations can be used to solve problems involving fractions. For example, if you have log⁡b(xy)=z\log_b(\frac{x}{y}) = z, you can use the properties of logarithms to simplify the expression.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a computer program. You can also use the properties of logarithms to simplify the function and make it easier to graph.

Q: What are some common mistakes to avoid when working with logarithmic equations?

A: Some common mistakes to avoid when working with logarithmic equations include:

  • Forgetting to check the domain: Make sure to check the domain of the logarithmic function before solving the equation.
  • Forgetting to check the range: Make sure to check the range of the logarithmic function before solving the equation.
  • Not using the properties of logarithms: Make sure to use the properties of logarithms to simplify the equation and make it easier to solve.

Conclusion

In this article, we have answered some frequently asked questions about logarithmic equations, providing a deeper understanding of the subject. We have covered topics such as the properties of logarithms, how to simplify logarithmic expressions, and how to solve logarithmic equations. We have also discussed some common applications of logarithmic equations and some common mistakes to avoid when working with them.

References