The given logarithmic expression is:

$\log\left(\sqrt{x^7 y^{-10}}\right)$

Step 1: Rewrite the square root

The square root can be expressed as an exponent of $\frac{1}{2}$:

$\sqrt{x^7 y^{-10}} = (x^7 y^{-10})^{\frac{1}{2}}$

Step 2: Apply the power rule

Using the logarithmic power rule $\log(a^b) = b\log(a)$, we get:

$\log\left((x^7 y^{-10})^{\frac{1}{2}}\right) = \frac{1}{2} \log(x^7 y^{-10})$

Step 3: Apply the product rule

Using the logarithmic property $\log(ab) = \log(a) + \log(b)$:

$\frac{1}{2} \left[ \log(x^7) + \log(y^{-10}) \right]$

Step 4: Apply the power rule separately

$\log(x^7) = 7\log(x), \quad \log(y^{-10}) = -10\log(y)$

Thus, the expression simplifies to:

$\frac{1}{2} \left( 7\log(x) - 10\log(y) \right)$

Final Answer:

$\frac{7}{2} \log(x) - 5\log(y)$

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Would you like more details or have any questions?

Here are some related questions to explore:

1. What are the properties of logarithms used in simplifying expressions?
2. How do logarithms help in solving exponential equations?
3. Can logarithms handle negative or zero values inside the argument?
4. How do you convert logarithmic expressions to exponential form?
5. What are the different bases of logarithms and their applications?

Tip: When simplifying logarithmic expressions, always check for opportunities to apply power, product, and quotient rules for simplification.