Which Of The Following Shows The Equation $\log _4(x+6)=2$ Rewritten Using Logarithms?A. $4^{\log _4(x+8)}=2^2$ B. $4^{\log _4(x+8)}=4^4$ C. $\log _4(x+6)=\log _4 16$ D. $\log _4(x+6)=\log _2 16$

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Introduction

Logarithmic equations are a fundamental concept in mathematics, and understanding how to rewrite them is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will focus on rewriting the equation log⁡4(x+6)=2\log _4(x+6)=2 using logarithms. We will explore the properties of logarithms, apply them to the given equation, and evaluate the possible answers.

Understanding Logarithms

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns the exponent to which a base number must be raised to produce the input number. In other words, if y=log⁡b(x)y = \log_b(x), then by=xb^y = x. The base of a logarithm is the number that is raised to the power of the exponent.

Rewriting the Equation

To rewrite the equation log⁡4(x+6)=2\log _4(x+6)=2 using logarithms, we need to apply the definition of a logarithm. We know that if y=log⁡b(x)y = \log_b(x), then by=xb^y = x. In this case, we have y=2y = 2 and b=4b = 4. Therefore, we can rewrite the equation as:

42=x+64^2 = x + 6

Evaluating the Possible Answers

Now that we have rewritten the equation, let's evaluate the possible answers:

A. 4log⁡4(x+8)=224^{\log _4(x+8)}=2^2

This option is incorrect because the base of the logarithm is 4, not 2. Additionally, the argument of the logarithm is x+8x+8, not x+6x+6.

B. 4log⁡4(x+8)=444^{\log _4(x+8)}=4^4

This option is incorrect because the argument of the logarithm is x+8x+8, not x+6x+6. Additionally, the exponent of the logarithm is 4, not 2.

C. log⁡4(x+6)=log⁡416\log _4(x+6)=\log _4 16

This option is correct because the base of the logarithm is 4, and the argument of the logarithm is x+6x+6. We can rewrite the equation as:

log⁡4(x+6)=log⁡416\log _4(x+6) = \log _4 16

x+6=16x + 6 = 16

x=10x = 10

D. log⁡4(x+6)=log⁡216\log _4(x+6)=\log _2 16

This option is incorrect because the base of the logarithm is 4, not 2. Additionally, the argument of the logarithm is x+6x+6, not 16.

Conclusion

In conclusion, the correct answer is C. log⁡4(x+6)=log⁡416\log _4(x+6)=\log _4 16. We applied the definition of a logarithm to rewrite the equation log⁡4(x+6)=2\log _4(x+6)=2 using logarithms. We evaluated the possible answers and found that option C is the only correct solution.

Properties of Logarithms

Logarithmic equations can be rewritten using various properties of logarithms. Some of the most common properties include:

  • Product Property: log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y
  • Quotient Property: log⁡b(xy)=log⁡bx−log⁡by\log_b(\frac{x}{y}) = \log_b x - \log_b y
  • Power Property: log⁡b(xy)=ylog⁡bx\log_b(x^y) = y\log_b x
  • Change of Base Property: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

These properties can be used to rewrite logarithmic equations in various ways, making it easier to solve them.

Examples of Rewriting Logarithmic Equations

Here are some examples of rewriting logarithmic equations using the properties of logarithms:

  • log⁡2(3x)=5\log_2(3x) = 5 can be rewritten as log⁡23+log⁡2x=5\log_2 3 + \log_2 x = 5
  • log⁡5(x2)=3\log_5(\frac{x}{2}) = 3 can be rewritten as log⁡5x−log⁡52=3\log_5 x - \log_5 2 = 3
  • log⁡3(x2)=4\log_3(x^2) = 4 can be rewritten as 2log⁡3x=42\log_3 x = 4
  • log⁡7x=log⁡2xlog⁡27\log_7 x = \frac{\log_2 x}{\log_2 7} can be rewritten as log⁡7x=log⁡2x⋅1log⁡27\log_7 x = \log_2 x \cdot \frac{1}{\log_2 7}

Tips for Rewriting Logarithmic Equations

Here are some tips for rewriting logarithmic equations:

  • Use the definition of a logarithm: If y=log⁡b(x)y = \log_b(x), then by=xb^y = x.
  • Apply the properties of logarithms: Use the product, quotient, power, and change of base properties to rewrite the equation.
  • Simplify the equation: Use algebraic manipulations to simplify the equation and make it easier to solve.
  • Check the answer: Verify that the rewritten equation is equivalent to the original equation.

Q: What is the definition of a logarithm?

A: A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns the exponent to which a base number must be raised to produce the input number. In other words, if y=log⁡b(x)y = \log_b(x), then by=xb^y = x.

Q: How do I rewrite a logarithmic equation using logarithms?

A: To rewrite a logarithmic equation using logarithms, you need to apply the definition of a logarithm. You can also use the properties of logarithms, such as the product, quotient, power, and change of base properties, to rewrite the equation.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

  • Product Property: log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y
  • Quotient Property: log⁡b(xy)=log⁡bx−log⁡by\log_b(\frac{x}{y}) = \log_b x - \log_b y
  • Power Property: log⁡b(xy)=ylog⁡bx\log_b(x^y) = y\log_b x
  • Change of Base Property: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

Q: How do I use the product property to rewrite a logarithmic equation?

A: To use the product property to rewrite a logarithmic equation, you need to multiply the arguments of the logarithms. For example, if you have log⁡2(3x)\log_2(3x), you can rewrite it as log⁡23+log⁡2x\log_2 3 + \log_2 x.

Q: How do I use the quotient property to rewrite a logarithmic equation?

A: To use the quotient property to rewrite a logarithmic equation, you need to divide the arguments of the logarithms. For example, if you have log⁡5(x2)\log_5(\frac{x}{2}), you can rewrite it as log⁡5x−log⁡52\log_5 x - \log_5 2.

Q: How do I use the power property to rewrite a logarithmic equation?

A: To use the power property to rewrite a logarithmic equation, you need to multiply the exponent by the logarithm of the base. For example, if you have log⁡3(x2)\log_3(x^2), you can rewrite it as 2log⁡3x2\log_3 x.

Q: How do I use the change of base property to rewrite a logarithmic equation?

A: To use the change of base property to rewrite a logarithmic equation, you need to change the base of the logarithm. For example, if you have log⁡7x\log_7 x, you can rewrite it as log⁡2xlog⁡27\frac{\log_2 x}{\log_2 7}.

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

A: Some common mistakes to avoid when rewriting logarithmic equations include:

  • Not using the definition of a logarithm: Make sure to apply the definition of a logarithm to rewrite the equation.
  • Not using the properties of logarithms: Make sure to use the product, quotient, power, and change of base properties to rewrite the equation.
  • Not simplifying the equation: Make sure to simplify the equation using algebraic manipulations.
  • Not checking the answer: Make sure to verify that the rewritten equation is equivalent to the original equation.

Q: How do I check if a rewritten logarithmic equation is equivalent to the original equation?

A: To check if a rewritten logarithmic equation is equivalent to the original equation, you need to verify that the two equations are equal. You can do this by substituting the rewritten equation into the original equation and checking if the two equations are equal.

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

A: Some real-world applications of rewriting logarithmic equations include:

  • Finance: Logarithmic equations are used to calculate interest rates and investment returns.
  • Science: Logarithmic equations are used to model population growth and decay.
  • Engineering: Logarithmic equations are used to design and optimize systems.
  • Computer Science: Logarithmic equations are used to analyze and optimize algorithms.

By understanding how to rewrite logarithmic equations, you can apply this knowledge to a wide range of real-world applications.