We are given that $\log_{10} 4 = 0.6021$ and asked to find $\log_{10} \sqrt{400}$ without using a calculator. Let's break down the problem step by step.

Step 1: Simplify $\log_{10} \sqrt{400}$

We can rewrite the square root using exponents:

$\sqrt{400} = 400^{1/2}$

Thus:

$\log_{10} \sqrt{400} = \log_{10} (400^{1/2}) = \frac{1}{2} \log_{10} 400$

Step 2: Simplify $\log_{10} 400$

We know that:

$400 = 4 \times 100$

So we can use the logarithmic property $\log(ab) = \log a + \log b$ to get:

$\log_{10} 400 = \log_{10} 4 + \log_{10} 100$

We already know that $\log_{10} 4 = 0.6021$, and since $\log_{10} 100 = 2$ (because $100 = 10^2$), we can substitute:

$\log_{10} 400 = 0.6021 + 2 = 2.6021$

Step 3: Compute $\frac{1}{2} \log_{10} 400$

Now, we need to divide the result by 2:

$\log_{10} \sqrt{400} = \frac{1}{2} \times 2.6021 = 1.30105$

Final Answer:

$\log_{10} \sqrt{400} \approx 1.3011$ (Rounding to four decimal places.)

Would you like more details or have any further questions?

Here are 5 related questions to explore this topic:

1. How would you compute $\log_{10} \sqrt{625}$ without a calculator?
2. What is the logarithmic property that allows us to simplify $\log_{10} \sqrt{400}$?
3. How can you express logarithms with base $e$ (natural logarithms)?
4. Can you use the same approach to solve $\log_{10} \sqrt{50}$?
5. How would this problem change if the logarithm was base 2 instead of base 10?

Tip: Remember that logarithmic properties, such as $\log(a \times b) = \log a + \log b$ and $\log(a^b) = b \log a$, are essential tools in simplifying expressions.