Fill in the blank(s) with the appropriate option.

Options

log 3600 can also be expressed as

(2 * log(6)) + 1

(6 * log(2)) + 1

(2 * log(6)) + 2

(6 * log(2)) + 2

Let's solve the problem step-by-step.

We need to express \( \log 3600 \) in terms of one of the provided options.

### Step 1: Prime factorization of 3600
We start by factoring 3600:
\[
3600 = 36 \times 100 = (6^2) \times (10^2) = 6^2 \times (2 \times 5)^2
\]
Thus, we have:
\[
3600 = 6^2 \times 2^2 \times 5^2
\]

### Step 2: Taking the logarithm
Now take the logarithm of both sides:
\[
\log 3600 = \log(6^2 \times 2^2 \times 5^2)
\]
Using the properties of logarithms, we can break this down:
\[
\log 3600 = \log(6^2) + \log(2^2) + \log(5^2)
\]
\[
\log 3600 = 2\log 6 + 2\log 2 + 2\log 5
\]

### Step 3: Simplifying further
Since \( \log 10 = \log(2 \times 5) = \log 2 + \log 5 \), we know that \( \log 10 = 1 \). Therefore, the sum \( \log 2 + \log 5 = 1 \), giving:
\[
2\log 2 + 2\log 5 = 2
\]

So, we now have:
\[
\log 3600 = 2\log 6 + 2
\]

### Conclusion:
The correct expression for \( \log 3600 \) is:
\[
\log 3600 = (2\log 6) + 2
\]

Thus, the correct option is:
\[
\boxed{(2\log 6) + 2}
\]

---

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you expand \( \log(AB) \) using logarithmic properties?
2. What is the value of \( \log 10 \) and why?
3. How does the property \( \log(a^b) = b \log a \) help in simplifying logarithms?
4. What is the difference between natural logarithms and common logarithms?
5. How would you express \( \log(360) \) using a similar method?

**Tip:** Remember to always use logarithmic properties like \( \log(ab) = \log a + \log b \) and \( \log(a^b) = b\log a \) to break down complex expressions easily.

Explore how to simplify the logarithmic expression log 3600 using prime factorization and logarithmic properties. This step-by-step solution covers key logarithmic concepts and formulas, suitable for students in Grades 10-12.

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