A Teacher Used The Change Of Base Formula To Determine Whether The Equation Below Is Correct:$ \left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3 $Which Statement Explains Whether The Equation Is Correct?A. The Equation Is

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Introduction

In mathematics, the change of base formula is a powerful tool used to simplify logarithmic expressions and solve equations involving different bases. A teacher, in an effort to assess their students' understanding of this concept, presented the following equation:

(log⁑210)(log⁑48)(log⁑104)=3\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3

The teacher asked the students to determine whether the equation is correct using the change of base formula. In this article, we will explore the solution to this problem and provide a clear explanation of the steps involved.

Understanding the Change of Base Formula

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

log⁑ba=log⁑calog⁑cb\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 enables us to change the base of a logarithm from bb to cc.

Applying the Change of Base Formula to the Given Equation

To verify the given equation, we need to apply the change of base formula to each of the logarithmic expressions. Let's start with the first expression:

log⁑210\log_2 10

Using the change of base formula, we can rewrite this expression as:

log⁑210=log⁑1010log⁑102\log_2 10 = \frac{\log_{10} 10}{\log_{10} 2}

Since log⁑1010=1\log_{10} 10 = 1, we can simplify this expression to:

log⁑210=1log⁑102\log_2 10 = \frac{1}{\log_{10} 2}

Simplifying the Second Expression

Next, we need to simplify the second expression:

log⁑48\log_4 8

Using the change of base formula, we can rewrite this expression as:

log⁑48=log⁑108log⁑104\log_4 8 = \frac{\log_{10} 8}{\log_{10} 4}

Since log⁑108=log⁑10(23)=3log⁑102\log_{10} 8 = \log_{10} (2^3) = 3\log_{10} 2, we can simplify this expression to:

log⁑48=3log⁑102log⁑104\log_4 8 = \frac{3\log_{10} 2}{\log_{10} 4}

Simplifying the Third Expression

Finally, we need to simplify the third expression:

log⁑104\log_{10} 4

Using the change of base formula, we can rewrite this expression as:

log⁑104=log⁑104log⁑1010\log_{10} 4 = \frac{\log_{10} 4}{\log_{10} 10}

Since log⁑104=log⁑10(22)=2log⁑102\log_{10} 4 = \log_{10} (2^2) = 2\log_{10} 2, we can simplify this expression to:

log⁑104=2log⁑102log⁑1010\log_{10} 4 = \frac{2\log_{10} 2}{\log_{10} 10}

Substituting the Simplified Expressions into the Original Equation

Now that we have simplified each of the logarithmic expressions, we can substitute them into the original equation:

(log⁑210)(log⁑48)(log⁑104)=3\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3

Substituting the simplified expressions, we get:

(1log⁑102)(3log⁑102log⁑104)(2log⁑102log⁑1010)=3\left(\frac{1}{\log_{10} 2}\right)\left(\frac{3\log_{10} 2}{\log_{10} 4}\right)\left(\frac{2\log_{10} 2}{\log_{10} 10}\right) = 3

Canceling Out Common Factors

We can simplify this expression by canceling out common factors. Notice that log⁑102\log_{10} 2 appears in both the numerator and denominator of the first two expressions. We can cancel out these factors to get:

(1log⁑102)(3log⁑102)(2log⁑102log⁑1010)=3\left(\frac{1}{\log_{10} 2}\right)\left(3\log_{10} 2\right)\left(\frac{2\log_{10} 2}{\log_{10} 10}\right) = 3

Simplifying the Expression Further

We can simplify this expression further by canceling out the log⁑102\log_{10} 2 terms:

3(2log⁑1010)=33\left(\frac{2}{\log_{10} 10}\right) = 3

Since log⁑1010=1\log_{10} 10 = 1, we can simplify this expression to:

3(2)=33(2) = 3

Conclusion

In conclusion, the equation (log⁑210)(log⁑48)(log⁑104)=3\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3 is not correct. The correct solution is:

3(2)=63(2) = 6

Therefore, the statement that explains whether the equation is correct is:

The equation is incorrect.

Final Answer

The final answer is: The equation is incorrect.

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Introduction

In our previous article, we explored the solution to the equation (log⁑210)(log⁑48)(log⁑104)=3\left(\log_2 10\right)\left(\log_4 8\right)\left(\log_{10} 4\right) = 3 using the change of base formula. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solution.

Q: What is the change of base formula?

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

log⁑ba=log⁑calog⁑cb\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: How do I apply the change of base formula to a logarithmic expression?

A: To apply the change of base formula, you need to identify the base of the logarithm and the value inside the logarithm. Then, you can rewrite the expression using the change of base formula:

log⁑ba=log⁑calog⁑cb\log_b a = \frac{\log_c a}{\log_c b}

Q: Can I use the change of base formula to simplify any logarithmic expression?

A: Yes, you can use the change of base formula to simplify any logarithmic expression. However, you need to be careful when choosing the new base, as it can affect the result.

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:

  • Not checking if the new base is valid (i.e., not equal to 1)
  • Not simplifying the expression correctly
  • Not canceling out common factors

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. However, you need to be careful when applying the formula and simplifying the expression.

Q: How do I know if the equation is correct or not?

A: To determine if the equation is correct or not, you need to apply the change of base formula and simplify the expression. If the result is equal to the given value, then the equation is correct. Otherwise, it is incorrect.

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 statistics and engineering

Q: Can I use the change of base formula to solve problems involving exponential functions?

A: Yes, you can use the change of base formula to solve problems involving exponential functions. However, you need to be careful when applying the formula and simplifying the expression.

Q: How do I choose the new base when using the change of base formula?

A: When choosing the new base, you need to consider the following factors:

  • The base of the original logarithm
  • The value inside the logarithm
  • The desired result

Q: Can I use the change of base formula to solve problems involving logarithmic inequalities?

A: Yes, you can use the change of base formula to solve problems involving logarithmic inequalities. However, you need to be careful when applying the formula and simplifying the expression.

Final Answer

The final answer is: The equation is incorrect.