If `1,log_(81)(3^(x)+48)" and "log_(9)(3^(x)-(8)/(3))` are in A.P., then find x

Options

A. 1
B. 2
C. 9
D. 3

Answer

To solve the problem, we need to find the value of \( x \) such that the logarithmic expressions \( 1, \log_{81}(3^x + 48) \), and \( \log_{9}(3^x - \frac{8}{3}) \) are in Arithmetic Progression (A.P.).

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
For three numbers \( a, b, c \) to be in A.P., the condition is:
\[ 2b = a + c \]
Here, let \( a = 1 \), \( b = \log_{81}(3^x + 48) \), and \( c = \log_{9}(3^x - \frac{8}{3}) \).

2. **Setting Up the Equation**:
From the A.P. condition, we have:
\[ 2 \log_{81}(3^x + 48) = 1 + \log_{9}(3^x - \frac{8}{3}) \]

3. **Changing the Base of Logarithms**:
We can convert the logarithms to a common base (base 3):
\[ \log_{81}(3^x + 48) = \frac{\log_{3}(3^x + 48)}{\log_{3}(81)} = \frac{\log_{3}(3^x + 48)}{4} \]
\[ \log_{9}(3^x - \frac{8}{3}) = \frac{\log_{3}(3^x - \frac{8}{3})}{\log_{3}(9)} = \frac{\log_{3}(3^x - \frac{8}{3})}{2} \]

4. **Substituting Back**:
Substitute these into the A.P. equation:
\[ 2 \cdot \frac{\log_{3}(3^x + 48)}{4} = 1 + \frac{\log_{3}(3^x - \frac{8}{3})}{2} \]
Simplifying gives:
\[ \frac{\log_{3}(3^x + 48)}{2} = 1 + \frac{\log_{3}(3^x - \frac{8}{3})}{2} \]

5. **Multiplying Through by 2**:
Multiply the entire equation by 2 to eliminate the fractions:
\[ \log_{3}(3^x + 48) = 2 + \log_{3}(3^x - \frac{8}{3}) \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ \log_{3}(3^x + 48) - \log_{3}(3^x - \frac{8}{3}) = 2 \]
Using the property of logarithms \( \log_{a}(b) - \log_{a}(c) = \log_{a}(\frac{b}{c}) \):
\[ \log_{3}\left(\frac{3^x + 48}{3^x - \frac{8}{3}}\right) = 2 \]

7. **Exponentiating**:
Exponentiate both sides to remove the logarithm:
\[ \frac{3^x + 48}{3^x - \frac{8}{3}} = 3^2 \]
Simplifying gives:
\[ \frac{3^x + 48}{3^x - \frac{8}{3}} = 9 \]

8. **Cross-Multiplying**:
Cross-multiplying results in:
\[ 3^x + 48 = 9(3^x - \frac{8}{3}) \]
Expanding the right side:
\[ 3^x + 48 = 9 \cdot 3^x - 24 \]

9. **Rearranging Terms**:
Rearranging gives:
\[ 48 + 24 = 9 \cdot 3^x - 3^x \]
\[ 72 = 8 \cdot 3^x \]

10. **Solving for \( x \)**:
Dividing both sides by 8:
\[ 3^x = 9 \]
Since \( 9 = 3^2 \):
\[ x = 2 \]

### Final Answer:
The value of \( x \) is \( 2 \).