31. 1 log y ^ xy + 1 log x ^ xy =
A)1

B)y

C)x

D)0

32. log x ^ sqrt x +log y ^ sqrt y^ 3 =

A)2

B)1

C)0

D)4

Let's solve both problems step by step.

### Problem 1:
\[
31. \ 1 \cdot \log(y^{xy}) + 1 \cdot \log(x^{xy}) = ?
\]
Using the logarithmic property:  
\[
\log(a^b) = b \cdot \log(a)
\]
We can rewrite the equation:

\[
1 \cdot \log(y^{xy}) + 1 \cdot \log(x^{xy}) = xy \cdot \log(y) + xy \cdot \log(x)
\]

Now factor out \(xy\):
\[
xy \cdot (\log(y) + \log(x))
\]

Using the property \(\log(a) + \log(b) = \log(ab)\), this becomes:
\[
xy \cdot \log(xy)
\]

So the entire expression simplifies to:
\[
xy \cdot \log(xy)
\]

Hence, the correct answer is **none of the given choices** (A-D), but if we had to guess based on likely exam errors, the **best match might be 1**. However, this depends on more context.

### Problem 2:
\[
32. \ \log(x^{\sqrt{x}}) + \log(y^{\sqrt{y^3}}) = ?
\]
Applying the logarithmic property again, \(\log(a^b) = b \cdot \log(a)\):

\[
\log(x^{\sqrt{x}}) = \sqrt{x} \cdot \log(x)
\]
\[
\log(y^{\sqrt{y^3}}) = \sqrt{y^3} \cdot \log(y) = y^{3/2} \cdot \log(y)
\]

Now we combine these:
\[
\sqrt{x} \cdot \log(x) + y^{3/2} \cdot \log(y)
\]

Since there's no further simplification possible, we cannot directly evaluate this to match any of the answer choices without specific values for \(x\) and \(y\). Given that no values are provided, **none of the options A-D fit directly**, but based on context the **answer closest to common scenarios could be 1 or 2**.

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Do you want to go over the details or have any questions?

---

Here are 5 related questions to deepen your understanding:

1. How do you use the logarithmic properties \(\log(a^b) = b \log(a)\)?
2. What are the common logarithmic simplifications when combining terms?
3. Can you explain how to factor out common terms in logarithmic expressions?
4. What are the most useful logarithmic identities to simplify expressions?
5. How do you approach problems with multiple logarithmic terms?

**Tip:** Always look for opportunities to use the properties of logarithms (product, quotient, and power rules) to simplify expressions easily.

31. 1 log y ^ xy + 1 log x ^ xy = A)1 B)y C)x D)0
32. log x ^ sqrt x + log y ^ sqrt y^ 3 = A)2 B)1 C)0 D)4

Explore step-by-step solutions to logarithmic problems involving exponents such as 'log y ^ xy + log x ^ xy' and 'log x ^ sqrt x + log y ^ sqrt y^3'. These problems use key algebraic concepts like the logarithmic power and product rules, suitable for Grades 10-12.

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