The value of `log_(9)27-log_(27)9` is

Options

A. `0`
B. `3//2`
C. `5//6`
D. none

Answer

To solve the expression \( \log_{9}27 - \log_{27}9 \), we can use properties of logarithms. Here’s a step-by-step solution:

### Step 1: Rewrite the logarithms in terms of a common base
We can express both logarithms using base 3, since both 9 and 27 are powers of 3:
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)

Thus, we can rewrite:
\[ \log_{9}27 = \log_{3^2}3^3 \]
\[ \log_{27}9 = \log_{3^3}3^2 \]

### Step 2: Apply the change of base formula
Using the change of base formula \( \log_{a}b = \frac{\log_{c}b}{\log_{c}a} \), we can rewrite the logarithms:
\[ \log_{9}27 = \frac{\log_{3}27}{\log_{3}9} = \frac{\log_{3}(3^3)}{\log_{3}(3^2)} = \frac{3}{2} \]
\[ \log_{27}9 = \frac{\log_{3}9}{\log_{3}27} = \frac{\log_{3}(3^2)}{\log_{3}(3^3)} = \frac{2}{3} \]

### Step 3: Substitute back into the expression
Now we substitute these values back into the original expression:
\[ \log_{9}27 - \log_{27}9 = \frac{3}{2} - \frac{2}{3} \]

### Step 4: Find a common denominator and simplify
The common denominator for 2 and 3 is 6. We convert each fraction:
\[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]
\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \]

Now we can perform the subtraction:
\[ \frac{9}{6} - \frac{4}{6} = \frac{9 - 4}{6} = \frac{5}{6} \]

### Final Answer
Thus, the value of \( \log_{9}27 - \log_{27}9 \) is:
\[ \frac{5}{6} \]

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