Supposed log5(6)=*a* and log5(11)=*b*. Use the change-of-base formula along with properties of logarithms to rewrite the expression in terms of *a* and *b*

. Show the steps for solving. log11(6)

We are tasked with expressing \( \log_{11}(6) \) in terms of \( a \) and \( b \), where \( \log_5(6) = a \) and \( \log_5(11) = b \). 

To solve this, we'll use the **change-of-base formula** and **logarithmic properties**. Here's a step-by-step breakdown:

### Step 1: Apply the Change-of-Base Formula

The change-of-base formula is:

\[
\log_c(x) = \frac{\log_b(x)}{\log_b(c)}
\]

In this case, we need to express \( \log_{11}(6) \) in terms of base 5, since we have information about logarithms in base 5. Using the change-of-base formula:

\[
\log_{11}(6) = \frac{\log_5(6)}{\log_5(11)}
\]

### Step 2: Substitute the Known Values

We are given that:

- \( \log_5(6) = a \)
- \( \log_5(11) = b \)

Substitute these into the equation:

\[
\log_{11}(6) = \frac{a}{b}
\]

### Final Answer:

Thus, \( \log_{11}(6) \) can be rewritten as:

\[
\log_{11}(6) = \frac{a}{b}
\]

### Explanation:

This result follows directly from the change-of-base formula and the properties of logarithms. By expressing \( \log_{11}(6) \) in terms of logarithms with base 5, we can utilize the values of \( a \) and \( b \) to represent the result.

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### Relative Questions:
1. How can we apply the change-of-base formula to other logarithmic bases?
2. What happens if we want to convert from base 10 to base 5 using the change-of-base formula?
3. How does the property \( \log_b(xy) = \log_b(x) + \log_b(y) \) affect solving such problems?
4. Can you prove the change-of-base formula using the definition of logarithms?
5. How can logarithmic properties simplify complex logarithmic expressions?

### Tip:
Whenever you need to change the base of a logarithm, the change-of-base formula is an essential tool. Keep it handy for converting between different bases easily!

Supposed log5(6)=a and log5(11)=b. Use the change-of-base formula along with properties of logarithms to rewrite the expression in terms of a and b. Show the steps for solving log11(6).

Learn how to express log11(6) in terms of a and b using the change-of-base formula and logarithmic properties. This step-by-step guide shows how log5(6) = a and log5(11) = b are applied. Suitable for students in Grades 10-12.

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