https://file.aiquickdraw.com/mathbot/file/1725816525480lrgm527z.jpg

The problem in the image reads:

"Jika log 2 = x dan log 3 = y maka nilai log 0,45 adalah ..."

This translates to:

"If log 2 = x and log 3 = y, then the value of log 0.45 is ..."

We need to express \(\log 0.45\) in terms of \(x\) and \(y\), using the properties of logarithms.

### Step-by-Step Solution:
1. **Breakdown of 0.45:**
   \[
   0.45 = \frac{9}{20}
   \]
   Since \(\log(a/b) = \log a - \log b\), we can rewrite:
   \[
   \log 0.45 = \log \left(\frac{9}{20}\right) = \log 9 - \log 20
   \]

2. **Simplify \(\log 9\):**
   \[
   9 = 3^2 \quad \text{so} \quad \log 9 = \log(3^2) = 2 \log 3 = 2y
   \]

3. **Simplify \(\log 20\):**
   \[
   20 = 2^2 \times 5 \quad \text{so} \quad \log 20 = \log(2^2 \times 5) = \log(2^2) + \log 5 = 2 \log 2 + \log 5 = 2x + \log 5
   \]
   We don't know \(\log 5\), but we can estimate it using the change of base rule:
   \[
   \log 5 \approx \log 10 - \log 2 = 1 - x
   \]
   Therefore, \(\log 20 = 2x + (1 - x) = x + 1\).

4. **Combine the results:**
   Now substitute back into the equation for \(\log 0.45\):
   \[
   \log 0.45 = 2y - (x + 1) = 2y - x - 1
   \]

Thus, the value of \(\log 0.45\) in terms of \(x\) and \(y\) is \(2y - x - 1\).

### Answer:
The correct option is **d. \(2y - x - 1\)**.

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Would you like a more detailed explanation or have any further questions?

Here are 5 related questions to enhance understanding:

1. How can you use logarithmic properties to simplify complex logarithmic expressions?
2. How would the approach change if you were given logarithms of other numbers (e.g., \(\log 5\) directly)?
3. How does the change of base rule help in estimating logarithmic values?
4. What is the significance of breaking down a number like 0.45 in terms of fractions for logarithmic calculations?
5. How does understanding the logarithmic expansion help in solving real-world problems?

**Tip:** When simplifying logarithmic expressions, always remember the key properties of logarithms such as the product, quotient, and power rules.

Jika log 2 = x dan log 3 = y maka nilai log 0,45 adalah?

Explore how to express log 0.45 in terms of x and y, given that log 2 = x and log 3 = y. This problem uses key logarithmic properties like the quotient and power rules, suitable for students in Grades 9-11.

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