Let's break down the problem.

We are asked to find which of the following is equal to:

$\log \sqrt[4]{1,000}$

We can rewrite this expression step by step:

1. Step 1: Simplify the radical expression: $\sqrt[4]{1,000} = 1,000^{1/4}$

2. Step 2: Apply the logarithm property: $\log(1,000^{1/4}) = \frac{1}{4} \log(1,000)$

3. Step 3: Simplify the logarithm: We know that $\log(1,000) = 3$ because $10^3 = 1,000$. $\log(1,000) = 3$

4. Step 4: Finish the expression: $\frac{1}{4} \log(1,000) = \frac{1}{4} \times 3 = \frac{3}{4}$

Thus, the correct answer is $\frac{3}{4}$.

Final Answer:

The correct option is the last one: $\boxed{\frac{3}{4}}$

Would you like further details on any of these steps or have any other questions?

Here are 5 related questions to explore:

1. How do you simplify a logarithmic expression with radicals?
2. What are the properties of logarithms used to solve such problems?
3. Can you generalize the solution if the base was not 10 but some other number?
4. How do you calculate logarithms for non-base 10 numbers?
5. What is the difference between $\log$ and $\ln$?

Tip: Logarithmic properties, such as $\log(a^b) = b \log(a)$, are essential for simplifying complex expressions.