Find the value of the following
`log_(9) 27`

Answer

To find the value of \( \log_{9} 27 \), we can follow these steps:

### Step 1: Rewrite the logarithm in terms of base 3
We know that both 9 and 27 can be expressed as powers of 3:
- \( 9 = 3^2 \)
- \( 27 = 3^3 \)

Using this, we can rewrite the logarithm:
\[ \log_{9} 27 = \log_{3^2} 3^3 \]

### Step 2: Use the change of base formula
Using the property of logarithms \( \log_{a^m} b^n = \frac{n}{m} \log_{a} b \), we can simplify:
\[ \log_{3^2} 3^3 = \frac{3}{2} \log_{3} 3 \]

### Step 3: Evaluate \( \log_{3} 3 \)
We know that \( \log_{a} a = 1 \) for any positive \( a \). Therefore:
\[ \log_{3} 3 = 1 \]

### Step 4: Substitute back into the equation
Now substituting back, we have:
\[ \log_{9} 27 = \frac{3}{2} \cdot 1 = \frac{3}{2} \]

### Final Answer
Thus, the value of \( \log_{9} 27 \) is:
\[ \frac{3}{2} \]

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