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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Would you like more details on this solution, or have any questions?

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!