The expression shown in the image is:

$\log_{\left(2 \sqrt{2}\right)}\left(\frac{1}{4}\right)$

Step-by-step Solution:

1. Base Simplification: The base of the logarithm is $2 \sqrt{2}$. We can express this in terms of powers of 2: $2 \sqrt{2} = 2 \cdot 2^{1/2} = 2^{3/2}$ So, the logarithmic expression becomes: $\log_{2^{3/2}}\left(\frac{1}{4}\right)$

2. Convert the Argument: We know that $\frac{1}{4} = 2^{-2}$, so we rewrite the expression as: $\log_{2^{3/2}}(2^{-2})$

3. Apply Logarithmic Property: The property of logarithms $\log_{a}(a^b) = b$ helps us here. Rewriting in simpler terms, we have: $\frac{\log_2(2^{-2})}{\log_2(2^{3/2})}$ Simplifying both terms:

   • $\log_2(2^{-2}) = -2$
   • $\log_2(2^{3/2}) = \frac{3}{2}$

   Therefore, the expression becomes: $\frac{-2}{3/2} = -2 \times \frac{2}{3} = -\frac{4}{3}$

Final Answer:

$\log_{\left(2 \sqrt{2}\right)}\left(\frac{1}{4}\right) = -\frac{4}{3}$

Would you like further explanation on any step or more details? Here are some follow-up questions for additional practice:

1. How do you simplify expressions like $2 \sqrt{2}$ in terms of powers of 2?
2. What is the general property of logarithms $\log_{a}(a^b)$ and how is it applied?
3. How would you solve $\log_{\left(3 \sqrt{3}\right)}(1/9)$?
4. What other properties of logarithms can simplify expressions like this one?
5. How does the logarithm change when the argument is a fraction?

Tip: Always try to express both the base and the argument in terms of the same number to make logarithmic calculations easier!