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.