If ` log_(10) 2 = a, log_(10)3 = b" then "log_(0.72)(9.6)` in terms of a and b is equal to

Options

A. `(2a+3b-1)/(5a+b-2)`
B. `(5a+b-1)/(3a+2b-2)`
C. `(3a+b-2)/(2a+3b-1)`
D. `(2a+5b-2)/(3a+b-1)`

Explanation

To solve the problem of finding \( \log_{0.72}(9.6) \) in terms of \( a \) and \( b \), where \( \log_{10} 2 = a \) and \( \log_{10} 3 = b \), we can follow these steps:

### Step 1: Change of Base Formula
We start by using the change of base formula for logarithms:
\[ \log_{0.72}(9.6) = \frac{\log_{10}(9.6)}{\log_{10}(0.72)} \]

### Step 2: Expressing 9.6 and 0.72
Next, we express \( 9.6 \) and \( 0.72 \) in terms of their prime factors:
- \( 9.6 = \frac{96}{10} = \frac{2^5 \cdot 3}{10} \)
- \( 0.72 = \frac{72}{100} = \frac{2^3 \cdot 3^2}{10^2} \)

### Step 3: Logarithmic Expressions
Now we can write the logarithms:
\[ \log_{10}(9.6) = \log_{10}\left(\frac{96}{10}\right) = \log_{10}(96) - \log_{10}(10) \]
\[ \log_{10}(0.72) = \log_{10}\left(\frac{72}{100}\right) = \log_{10}(72) - \log_{10}(100) \]

### Step 4: Simplifying Logarithms
Using the properties of logarithms:
\[ \log_{10}(10) = 1 \quad \text{and} \quad \log_{10}(100) = 2 \]
Thus, we have:
\[ \log_{10}(9.6) = \log_{10}(96) - 1 \]
\[ \log_{10}(0.72) = \log_{10}(72) - 2 \]

### Step 5: Factorizing 96 and 72
Next, we factor \( 96 \) and \( 72 \):
- \( 96 = 2^5 \cdot 3 \)
- \( 72 = 2^3 \cdot 3^2 \)

Now we can express the logarithms:
\[ \log_{10}(96) = \log_{10}(2^5) + \log_{10}(3) = 5\log_{10}(2) + \log_{10}(3) = 5a + b \]
\[ \log_{10}(72) = \log_{10}(2^3) + \log_{10}(3^2) = 3\log_{10}(2) + 2\log_{10}(3) = 3a + 2b \]

### Step 6: Substituting Back
Now we substitute back into our expressions for \( \log_{10}(9.6) \) and \( \log_{10}(0.72) \):
\[ \log_{10}(9.6) = (5a + b) - 1 = 5a + b - 1 \]
\[ \log_{10}(0.72) = (3a + 2b) - 2 = 3a + 2b - 2 \]

### Step 7: Final Expression
Now we can substitute these into our change of base formula:
\[ \log_{0.72}(9.6) = \frac{5a + b - 1}{3a + 2b - 2} \]

### Conclusion
Thus, the final answer is:
\[ \log_{0.72}(9.6) = \frac{5a + b - 1}{3a + 2b - 2} \]