https://file.aiquickdraw.com/mathbot/file/1727016084745jbu25hx8.jpg

The problem asks for the value of:

\[
{}^2\log 20 + {}^2\log 45 - {}^2\log 180 - {}^2\log 40
\]

We can simplify this expression using the properties of logarithms. Recall that:

1. \(\log_b (x \cdot y) = \log_b x + \log_b y\) (product rule).
2. \(\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y\) (quotient rule).

### Step 1: Combine the logs
Using the quotient rule for logarithms:

\[
{}^2\log 20 + {}^2\log 45 - {}^2\log 180 - {}^2\log 40
= {}^2\log \left( \frac{20 \cdot 45}{180 \cdot 40} \right)
\]

### Step 2: Simplify the fraction
First, compute the product in the numerator and denominator:

\[
\frac{20 \cdot 45}{180 \cdot 40} = \frac{900}{7200} = \frac{1}{8}
\]

### Step 3: Apply the logarithm
Now we apply the logarithm to the simplified fraction:

\[
{}^2\log \left(\frac{1}{8}\right)
\]

We know that \(\frac{1}{8} = 2^{-3}\), so:

\[
{}^2\log (2^{-3}) = -3
\]

Thus, the value of the expression is \(-3\).

The correct answer is **B. -3**.

---

Would you like more details or have any questions? Here are some related questions to further your understanding:

1. How do logarithmic properties help simplify complex expressions?
2. What is the general rule for simplifying logarithms of fractions?
3. How would the answer change if one of the logarithms had a different base?
4. Can you apply the same properties for natural logarithms (base \(e\))?
5. How can you express \(\log_b (x^n)\) using logarithmic properties?

**Tip**: Always look for opportunities to apply logarithmic properties (such as product, quotient, and power rules) to simplify expressions before calculating.

Nilai dari 2log 20 + 2log45 - 2log180 - 2log40 adalah . . .

Learn how to simplify the expression 2log 20 + 2log45 - 2log180 - 2log40 by applying logarithmic properties such as the quotient and product rules. This problem is suitable for students in Grades 10-12 and involves the key concepts of algebra and logarithmic manipulation.

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