Which Expression Results When The Change Of Base Formula Is Applied To $\log_4(x+2$\]?A. $\frac{\log (x+2)}{\log 4}$ B. $\frac{\log 4}{\log (x+2)}$ C. $\frac{\log 4}{\log X+2}$ D. $\frac{\log X+2}{\log 4}$

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The Change of Base Formula: A Mathematical Concept

The change of base formula is a fundamental concept in mathematics, particularly in the field of logarithms. It is a powerful tool that allows us to express a logarithm in terms of another base, making it easier to work with and manipulate. In this article, we will explore the change of base formula and apply it to a specific expression to determine the resulting expression.

Understanding the Change of Base Formula

The change of base formula is given by:

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

where aa, bb, and cc are positive real numbers, and c≠1c \neq 1. This formula allows us to express a logarithm in base bb in terms of a logarithm in base cc.

Applying the Change of Base Formula

Now, let's apply the change of base formula to the expression log⁑4(x+2)\log_4(x+2). We want to express this logarithm in terms of a different base. Let's choose base 10 as our new base.

Using the change of base formula, we get:

log⁑4(x+2)=log⁑10(x+2)log⁑10(4)\log_4(x+2) = \frac{\log_{10}(x+2)}{\log_{10}(4)}

Evaluating the Expression

Now, let's evaluate the expression log⁑10(x+2)log⁑10(4)\frac{\log_{10}(x+2)}{\log_{10}(4)}. We can simplify this expression by using the properties of logarithms.

Since log⁑10(4)=log⁑10(22)=2log⁑10(2)\log_{10}(4) = \log_{10}(2^2) = 2\log_{10}(2), we can rewrite the expression as:

log⁑10(x+2)log⁑10(4)=log⁑10(x+2)2log⁑10(2)\frac{\log_{10}(x+2)}{\log_{10}(4)} = \frac{\log_{10}(x+2)}{2\log_{10}(2)}

Simplifying the Expression

Now, let's simplify the expression further. We can use the property of logarithms that states log⁑a(bc)=clog⁑a(b)\log_a(b^c) = c\log_a(b).

Using this property, we can rewrite the expression as:

log⁑10(x+2)2log⁑10(2)=log⁑10(x+2)log⁑10(22)\frac{\log_{10}(x+2)}{2\log_{10}(2)} = \frac{\log_{10}(x+2)}{\log_{10}(2^2)}

Conclusion

In conclusion, the change of base formula is a powerful tool that allows us to express a logarithm in terms of another base. By applying the change of base formula to the expression log⁑4(x+2)\log_4(x+2), we get the resulting expression log⁑10(x+2)log⁑10(4)\frac{\log_{10}(x+2)}{\log_{10}(4)}. This expression can be simplified further using the properties of logarithms.

The Correct Answer

Based on our analysis, the correct answer is:

A. log⁑(x+2)log⁑4\frac{\log (x+2)}{\log 4}

This answer is consistent with the change of base formula and the properties of logarithms.

Comparison with Other Options

Let's compare our answer with the other options:

  • Option B: log⁑4log⁑(x+2)\frac{\log 4}{\log (x+2)} is incorrect because it reverses the order of the logarithms.
  • Option C: log⁑4log⁑x+2\frac{\log 4}{\log x+2} is incorrect because it uses a logarithm with a negative exponent.
  • Option D: log⁑x+2log⁑4\frac{\log x+2}{\log 4} is incorrect because it uses a logarithm with a negative exponent.

Conclusion

In conclusion, the change of base formula is a powerful tool that allows us to express a logarithm in terms of another base. By applying the change of base formula to the expression log⁑4(x+2)\log_4(x+2), we get the resulting expression log⁑10(x+2)log⁑10(4)\frac{\log_{10}(x+2)}{\log_{10}(4)}. This expression can be simplified further using the properties of logarithms. The correct answer is A. log⁑(x+2)log⁑4\frac{\log (x+2)}{\log 4}.

Final Answer

The final answer is A. log⁑(x+2)log⁑4\frac{\log (x+2)}{\log 4}.
Q&A: The Change of Base Formula

In our previous article, we explored the change of base formula and applied it to a specific expression to determine the resulting expression. In this article, we will answer some frequently asked questions about the change of base formula.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to express a logarithm in terms of another base. It is given by:

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

where aa, bb, and cc are positive real numbers, and c≠1c \neq 1.

Q: Why do we need the change of base formula?

A: We need the change of base formula because it allows us to express a logarithm in terms of a different base. This is useful when we need to work with logarithms in different bases.

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

A: To apply the change of base formula, you need to follow these steps:

  1. Identify the base of the logarithm you want to change.
  2. Choose a new base for the logarithm.
  3. Use the change of base formula to express the logarithm in the new base.

Q: What are some common mistakes to avoid when using the change of base formula?

A: Some common mistakes to avoid when using the change of base formula include:

  • Reversing the order of the logarithms.
  • Using a logarithm with a negative exponent.
  • Not simplifying the expression after applying the change of base formula.

Q: Can I use the change of base formula with any base?

A: Yes, you can use the change of base formula with any base, as long as the base is a positive real number and is not equal to 1.

Q: How do I simplify the expression after applying the change of base formula?

A: To simplify the expression after applying the change of base formula, you can use the properties of logarithms, such as the product rule and the quotient rule.

Q: Can I use the change of base formula to solve equations involving logarithms?

A: Yes, you can use the change of base formula to solve equations involving logarithms. By applying the change of base formula, you can express the logarithm in a different base, making it easier to solve the equation.

Q: What are some real-world applications of the change of base formula?

A: The change of base formula has many real-world applications, including:

  • Calculating the pH of a solution.
  • Determining the concentration of a solution.
  • Analyzing data in different bases.

Conclusion

In conclusion, the change of base formula is a powerful tool that allows us to express a logarithm in terms of another base. By understanding the change of base formula and how to apply it, you can solve a wide range of problems involving logarithms.

Frequently Asked Questions

Here are some frequently asked questions about the change of base formula:

  • Q: What is the change of base formula? A: The change of base formula is a mathematical formula that allows us to express a logarithm in terms of another base.
  • Q: Why do we need the change of base formula? A: We need the change of base formula because it allows us to express a logarithm in terms of a different base.
  • Q: How do I apply the change of base formula? A: To apply the change of base formula, you need to follow these steps: 1. Identify the base of the logarithm you want to change. 2. Choose a new base for the logarithm. 3. Use the change of base formula to express the logarithm in the new base.

Additional Resources

If you want to learn more about the change of base formula, here are some additional resources:

  • Khan Academy: Change of Base Formula
  • Mathway: Change of Base Formula
  • Wolfram Alpha: Change of Base Formula

Conclusion

In conclusion, the change of base formula is a powerful tool that allows us to express a logarithm in terms of another base. By understanding the change of base formula and how to apply it, you can solve a wide range of problems involving logarithms.