Tyler Applied The Change Of Base Formula To A Logarithmic Expression. The Resulting Expression Is Shown Below:$\[ \frac{\log \frac{1}{4}}{\log 12} \\]Which Expression Could Be Tyler's Original Expression?A. \[$\log _{\frac{1}{4}}

Tyler's Logarithmic Expression: A Change of Base Formula

Understanding the Change of Base Formula

The change of base formula is a fundamental concept in mathematics, particularly in the field of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. The formula is given by:

$$\log_b a = \frac{\log_c a}{\log_c b}$$

where $a$ and $b$ are positive real numbers, and $c$ is the new base.

Tyler's Original Expression

Tyler applied the change of base formula to a logarithmic expression, resulting in the expression:

$$\frac{\log \frac{1}{4}}{\log 12}$$

To find Tyler's original expression, we need to work backwards and apply the change of base formula in reverse.

Step 1: Simplify the Expression

Let's start by simplifying the expression:

$$\frac{\log \frac{1}{4}}{\log 12}$$

We can rewrite $\frac{1}{4}$ as $4^{-1}$, and then apply the property of logarithms that states $\log a^b = b \log a$:

$$\frac{\log 4^{-1}}{\log 12}$$

$$\frac{-1 \log 4}{\log 12}$$

Step 2: Apply the Change of Base Formula

Now, we can apply the change of base formula in reverse to rewrite the expression in terms of a different base. Let's choose base 4 as the new base:

$$\frac{-1 \log 4}{\log 12} = \frac{\log_4 4^{-1}}{\log_4 12}$$

Step 3: Simplify the Expression

Now, we can simplify the expression further:

$$\frac{\log_4 4^{-1}}{\log_4 12} = \frac{\log_4 \frac{1}{4}}{\log_4 12}$$

We can rewrite $\frac{1}{4}$ as $4^{-1}$, and then apply the property of logarithms that states $\log a^b = b \log a$:

$$\frac{\log_4 4^{-1}}{\log_4 12} = \frac{-1 \log_4 4}{\log_4 12}$$

Step 4: Simplify the Expression

Now, we can simplify the expression further:

$$\frac{-1 \log_4 4}{\log_4 12} = \frac{-1 \cdot 1}{\log_4 12}$$

$$\frac{-1}{\log_4 12}$$

Conclusion

Tyler's original expression is:

$$\log_{\frac{1}{4}} 12$$

This is the expression that Tyler would have obtained by applying the change of base formula in reverse.

Discussion

The change of base formula is a powerful tool in mathematics, particularly in the field of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. By applying the change of base formula in reverse, we can find the original expression that Tyler would have obtained.

Key Takeaways

• The change of base formula is given by: $\log_b a = \frac{\log_c a}{\log_c b}$
• To find Tyler's original expression, we need to apply the change of base formula in reverse.
• The original expression is: $\log_{\frac{1}{4}} 12$

References

• [1] "Change of Base Formula" by Math Open Reference
• [2] "Logarithms" by Khan Academy

Related Topics

• Logarithmic expressions
• Change of base formula
• Logarithmic properties

Further Reading

• "Logarithmic Expressions" by Math Is Fun
• "Change of Base Formula" by Wolfram MathWorld

Practice Problems

• Find the original expression that would result in the expression: $\frac{\log \frac{1}{16}}{\log 8}$
• Find the original expression that would result in the expression: $\frac{\log \frac{1}{9}}{\log 27}$
  Tyler's Logarithmic Expression: A Change of Base Formula - Q&A

Understanding the Change of Base Formula

The change of base formula is a fundamental concept in mathematics, particularly in the field of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. The formula is given by:

$$\log_b a = \frac{\log_c a}{\log_c b}$$

where $a$ and $b$ are positive real numbers, and $c$ is the new base.

Q&A

Q: What is the change of base formula?

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

$$\log_b a = \frac{\log_c a}{\log_c b}$$

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 and the argument of the logarithmic expression.
2. Choose a new base.
3. Apply the change of base formula using the new base.

Q: What is the original expression that would result in the expression: $\frac{\log \frac{1}{16}}{\log 8}$?

A: To find the original expression, we need to apply the change of base formula in reverse. Let's start by simplifying the expression:

$$\frac{\log \frac{1}{16}}{\log 8}$$

We can rewrite $\frac{1}{16}$ as $16^{-1}$, and then apply the property of logarithms that states $\log a^b = b \log a$:

$$\frac{\log 16^{-1}}{\log 8}$$

$$\frac{-1 \log 16}{\log 8}$$

Now, we can apply the change of base formula in reverse to rewrite the expression in terms of a different base. Let's choose base 2 as the new base:

$$\frac{-1 \log 16}{\log 8} = \frac{\log_2 16^{-1}}{\log_2 8}$$

We can rewrite $16^{-1}$ as $2^{-4}$, and then apply the property of logarithms that states $\log a^b = b \log a$:

$$\frac{\log_2 16^{-1}}{\log_2 8} = \frac{\log_2 2^{-4}}{\log_2 8}$$

Now, we can simplify the expression further:

$$\frac{\log_2 2^{-4}}{\log_2 8} = \frac{-4 \log_2 2}{\log_2 8}$$

$$\frac{-4 \cdot 1}{\log_2 8}$$

$$\frac{-4}{\log_2 8}$$

The original expression is:

$$\log_{16} 8$$

Q: What is the original expression that would result in the expression: $\frac{\log \frac{1}{9}}{\log 27}$?

A: To find the original expression, we need to apply the change of base formula in reverse. Let's start by simplifying the expression:

$$\frac{\log \frac{1}{9}}{\log 27}$$

We can rewrite $\frac{1}{9}$ as $9^{-1}$, and then apply the property of logarithms that states $\log a^b = b \log a$:

$$\frac{\log 9^{-1}}{\log 27}$$

$$\frac{-1 \log 9}{\log 27}$$

Now, we can apply the change of base formula in reverse to rewrite the expression in terms of a different base. Let's choose base 3 as the new base:

$$\frac{-1 \log 9}{\log 27} = \frac{\log_3 9^{-1}}{\log_3 27}$$

We can rewrite $9^{-1}$ as $3^{-2}$, and then apply the property of logarithms that states $\log a^b = b \log a$:

$$\frac{\log_3 9^{-1}}{\log_3 27} = \frac{\log_3 3^{-2}}{\log_3 27}$$

Now, we can simplify the expression further:

$$\frac{\log_3 3^{-2}}{\log_3 27} = \frac{-2 \log_3 3}{\log_3 27}$$

$$\frac{-2 \cdot 1}{\log_3 27}$$

$$\frac{-2}{\log_3 27}$$

The original expression is:

$$\log_{9} 27$$

Conclusion

The change of base formula is a powerful tool in mathematics, particularly in the field of logarithms. By applying the change of base formula in reverse, we can find the original expression that would result in a given expression. In this article, we have seen how to apply the change of base formula in reverse to find the original expression that would result in the expressions: $\frac{\log \frac{1}{16}}{\log 8}$ and $\frac{\log \frac{1}{9}}{\log 27}$.

Key Takeaways

• The change of base formula is given by: $\log_b a = \frac{\log_c a}{\log_c b}$
• To find the original expression that would result in a given expression, we need to apply the change of base formula in reverse.
• The original expression is: $\log_{16} 8$ and $\log_{9} 27$

References

• [1] "Change of Base Formula" by Math Open Reference
• [2] "Logarithms" by Khan Academy

Related Topics

• Logarithmic expressions
• Change of base formula
• Logarithmic properties

Further Reading

• "Logarithmic Expressions" by Math Is Fun
• "Change of Base Formula" by Wolfram MathWorld

Practice Problems

• Find the original expression that would result in the expression: $\frac{\log \frac{1}{25}}{\log 125}$
• Find the original expression that would result in the expression: $\frac{\log \frac{1}{8}}{\log 64}$