https://file.aiquickdraw.com/mathbot/file/17297860560998z59zntt.jpg

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!

Step-by-step solution for the logarithmic expression log(2√2)(1/4). Learn how to simplify the base and apply logarithmic properties to find the exact value. Ideal for students in Grades 9-12.

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