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. $2y - x - 1$.

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.