Evaluate: `log_(9)27 - log_(27)9`

Options

A. 0
B. `(3)/(2)`
C. `(5)/(6)`
D. `(2)/(3)`

Answer

To evaluate the expression \( \log_{9}27 - \log_{27}9 \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of base 3.
We know that:
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)

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

### Step 2: Use the change of base formula.
Using the change of base formula, we can express these logarithms as:
\[ \log_{a^b}c^d = \frac{d}{b} \log_{a}c \]
Applying this to our expressions:
\[ \log_{9}27 = \frac{3}{2} \log_{3}3 \]
\[ \log_{27}9 = \frac{2}{3} \log_{3}3 \]

### Step 3: Simplify the logarithms.
Since \( \log_{3}3 = 1 \), we can simplify:
\[ \log_{9}27 = \frac{3}{2} \cdot 1 = \frac{3}{2} \]
\[ \log_{27}9 = \frac{2}{3} \cdot 1 = \frac{2}{3} \]

### Step 4: Substitute back into the original expression.
Now we substitute these values back into the expression:
\[ \log_{9}27 - \log_{27}9 = \frac{3}{2} - \frac{2}{3} \]

### Step 5: Find a common denominator and simplify.
The common denominator for \( \frac{3}{2} \) and \( \frac{2}{3} \) is 6. We convert both fractions:
\[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]
\[ \frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \]

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

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

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