Simplify 2log $\frac{2}{5}$ - log$\frac{72}{125}$ + log 9

To simplify the expression $2\log(\frac{2}{5}) - \log(\frac{72}{125}) + \log(9)$, we will use the fundamental laws of logarithms.

1. Analyze the expression

The expression is: $2\log(\frac{2}{5}) - \log(\frac{72}{125}) + \log(9)$

2. Explain the concepts (Logarithm Laws)

• Power Rule: $n\log(x) = \log(x^n)$
• Subtraction Rule: $\log(a) - \log(b) = \log(\frac{a}{b})$
• Addition Rule: $\log(a) + \log(b) = \log(a \times b)$

3. Step-by-step Solution

Step 1: Apply the Power Rule to the first term $2\log(\frac{2}{5}) = \log((\frac{2}{5})^2) = \log(\frac{4}{25})$

Now the expression is: $\log(\frac{4}{25}) - \log(\frac{72}{125}) + \log(9)$

Step 2: Combine the first two terms using the Subtraction Rule $\log(\frac{4}{25}) - \log(\frac{72}{125}) = \log(\frac{4/25}{72/125})$ $= \log(\frac{4}{25} \times \frac{125}{72})$

Simplify the fraction: $= \log(\frac{1}{1} \times \frac{5}{18}) = \log(\frac{5}{18})$

Step 3: Add the final term using the Addition Rule $\log(\frac{5}{18}) + \log(9) = \log(\frac{5}{18} \times 9)$ $= \log(\frac{5}{2}) = \log(5) - \log(2)$

Wait! Let's re-examine the options. Since the options involve $\log(2)$ and $\log(3)$, let's express the numbers in terms of their prime factors (2, 3, and 5).

$\log(\frac{4}{25}) - \log(\frac{72}{125}) + \log(9)$ $= (\log 4 - \log 25) - (\log 72 - \log 125) + \log 9$ $= (2\log 2 - 2\log 5) - (3\log 2 + 2\log 3 - 3\log 5) + 2\log 3$ $= 2\log 2 - 2\log 5 - 3\log 2 - 2\log 3 + 3\log 5 + 2\log 3$

Step 4: Group like terms $= (2\log 2 - 3\log 2) + (-2\log 5 + 3\log 5) + (-2\log 3 + 2\log 3)$ $= -\log 2 + \log 5$

Since $\log 5 = \log(\frac{10}{2}) = \log 10 - \log 2 = 1 - \log 2$: $= -\log 2 + (1 - \log 2)$ $= 1 - 2\log 2$

4. Identify the answer

Comparing our result ($1 - 2\log 2$) to the given options, it matches option D.

5. Exam Tip

When you see logarithms with different bases or fractions, always try to break the numbers down into their prime factors (2, 3, 5). This makes it much easier to cancel out terms and reach the simplified form required by the options!