Given the equations:

$\log_a 5 = x \quad \text{and} \quad \log_a 2 = y,$

we need to find the value of:

$\log_2 \left(\frac{1}{5}\right).$

Step 1: Express $\frac{1}{5}$ in terms of logarithms

The expression can be rewritten using the properties of logarithms:

$\log_2 \left(\frac{1}{5}\right) = \log_2 (5^{-1}).$

Using the logarithmic property $\log_b (x^n) = n \log_b x$, this becomes:

$\log_2 \left(\frac{1}{5}\right) = -\log_2 5.$

Step 2: Convert $\log_2 5$ to base $a$

We can express $\log_2 5$ using the change of base formula:

$\log_2 5 = \frac{\log_a 5}{\log_a 2} = \frac{x}{y}.$

Thus:

$\log_2 \left(\frac{1}{5}\right) = -\frac{x}{y}.$

Final Answer:

$\log_2 \left(\frac{1}{5}\right) = -\frac{x}{y}.$

Would you like any further details or explanations?

Here are 8 related questions you might find interesting:

1. How would you solve for $\log_a 10$ using $x$ and $y$?
2. Can you express $\log_a \left(\frac{10}{5}\right)$ using $x$ and $y$?
3. What is the value of $\log_5 2$ in terms of $x$ and $y$?
4. How does the change of base formula help in logarithmic conversions?
5. What would $\log_5 (2^2)$ equal in terms of $x$ and $y$?
6. How would you calculate $\log_{10} 5$ using $x$ and $y$?
7. What are the logarithmic properties used in the calculation above?
8. How can you express $\log_{2a} 5$ in terms of $x$ and $y$?

Tip: When converting logarithms between different bases, the change of base formula is a powerful tool to simplify expressions.