The given expression is:

\[
\log \sqrt{a^b} \times \log \sqrt{b^c} \times \log \sqrt{c^a}
\]

We can simplify this step by step.

### Step 1: Use the square root property
Recall that \(\sqrt{x} = x^{1/2}\). So, each square root can be rewritten as a power of 1/2.

\[
\log \sqrt{a^b} = \log \left( a^b \right)^{1/2} = \frac{1}{2} \log (a^b)
\]
\[
\log \sqrt{b^c} = \frac{1}{2} \log (b^c)
\]
\[
\log \sqrt{c^a} = \frac{1}{2} \log (c^a)
\]

Now the expression becomes:

\[
\left( \frac{1}{2} \log (a^b) \right) \times \left( \frac{1}{2} \log (b^c) \right) \times \left( \frac{1}{2} \log (c^a) \right)
\]

### Step 2: Apply the logarithm power rule
The logarithm power rule states that \(\log(x^y) = y \log(x)\).

Thus:

\[
\log (a^b) = b \log a, \quad \log (b^c) = c \log b, \quad \log (c^a) = a \log c
\]

Substitute these into the expression:

\[
\left( \frac{1}{2} b \log a \right) \times \left( \frac{1}{2} c \log b \right) \times \left( \frac{1}{2} a \log c \right)
\]

### Step 3: Multiply the terms
Combine all the terms:

\[
\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}
\]
and
\[
b \log a \times c \log b \times a \log c = abc (\log a)(\log b)(\log c)
\]

Thus, the entire expression simplifies to:

\[
\frac{abc (\log a)(\log b)(\log c)}{8}
\]

### Final Answer:
The simplified form of the given expression is:

\[
\frac{abc (\log a)(\log b)(\log c)}{8}
\]

Would you like a detailed explanation of any specific part?

### 5 Related Questions:
1. How does the logarithmic power rule work?
2. Can logarithms of different bases be simplified further?
3. How would this expression change with different values for \(a\), \(b\), and \(c\)?
4. What are the properties of logarithms used in simplifications?
5. How can you apply logarithmic identities to more complex problems?

### Tip:
When simplifying logarithmic expressions, always look for opportunities to apply power and product rules to make calculations easier.

Simplify the expression log√a^b × log√b^c × log√c^a using properties of logarithms and exponents. Step-by-step breakdown includes applying the square root property and logarithmic power rule. Ideal for students in Grades 10-12.

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