What is `log_(81) 243` equal to ?

Options

A. 0.75
B. 1.25
C. 1.5
D. 3

Answer

To solve the problem \( \log_{81} 243 \), we can follow these steps:

### Step 1: Rewrite the logarithm
We start with the expression:
\[ \log_{81} 243 \]

### Step 2: Express 243 in terms of powers of 3
We know that:
\[ 243 = 3^5 \]
Thus, we can rewrite the logarithm as:
\[ \log_{81} (3^5) \]

### Step 3: Use the power rule of logarithms
According to the power rule of logarithms, \( \log_b (a^n) = n \cdot \log_b (a) \). Applying this rule, we get:
\[ \log_{81} (3^5) = 5 \cdot \log_{81} (3) \]

### Step 4: Express 81 in terms of powers of 3
Next, we note that:
\[ 81 = 3^4 \]
So we can rewrite the logarithm again:
\[ \log_{81} (3) = \log_{3^4} (3) \]

### Step 5: Use the change of base formula
Using the change of base formula \( \log_{a^n}(b) = \frac{1}{n} \log_a(b) \), we can simplify:
\[ \log_{3^4} (3) = \frac{1}{4} \log_{3} (3) \]

### Step 6: Evaluate \( \log_{3} (3) \)
Since \( \log_{3} (3) = 1 \):
\[ \log_{3^4} (3) = \frac{1}{4} \cdot 1 = \frac{1}{4} \]

### Step 7: Substitute back into the equation
Now we substitute back into our earlier expression:
\[ 5 \cdot \log_{81} (3) = 5 \cdot \frac{1}{4} = \frac{5}{4} \]

### Step 8: Convert to decimal
Finally, converting \( \frac{5}{4} \) to decimal gives us:
\[ \frac{5}{4} = 1.25 \]

### Final Answer
Thus, the value of \( \log_{81} 243 \) is:
\[ \boxed{1.25} \]