The value of x satisfying
`log_(3)4 -2 log_(3)sqrt(3x +1) =1 - log_(3)(5x -2)`

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To solve the equation

\[ \log_{3}4 - 2 \log_{3}\sqrt{3x + 1} = 1 - \log_{3}(5x - 2) \]

we will follow these steps:

### Step 1: Simplify the Left Side
Using the property of logarithms that states \( \log_{a}(b^{c}) = c \cdot \log_{a}(b) \), we can rewrite \( 2 \log_{3}\sqrt{3x + 1} \) as:

\[ 2 \log_{3}\sqrt{3x + 1} = \log_{3}((\sqrt{3x + 1})^{2}) = \log_{3}(3x + 1) \]

Thus, the left side becomes:

\[ \log_{3}4 - \log_{3}(3x + 1) \]

Using the property \( \log_{a}(b) - \log_{a}(c) = \log_{a}\left(\frac{b}{c}\right) \), we can combine the logarithms:

\[ \log_{3}\left(\frac{4}{3x + 1}\right) \]

### Step 2: Simplify the Right Side
The right side can be rewritten using the property of logarithms:

\[ 1 - \log_{3}(5x - 2) = \log_{3}(3) - \log_{3}(5x - 2) \]

Again, using the property \( \log_{a}(b) - \log_{a}(c) = \log_{a}\left(\frac{b}{c}\right) \):

\[ \log_{3}\left(\frac{3}{5x - 2}\right) \]

### Step 3: Set the Two Sides Equal
Now we have:

\[ \log_{3}\left(\frac{4}{3x + 1}\right) = \log_{3}\left(\frac{3}{5x - 2}\right) \]

Since the logarithms are equal, we can set their arguments equal to each other:

\[ \frac{4}{3x + 1} = \frac{3}{5x - 2} \]

### Step 4: Cross Multiply
Cross multiplying gives us:

\[ 4(5x - 2) = 3(3x + 1) \]

Expanding both sides:

\[ 20x - 8 = 9x + 3 \]

### Step 5: Solve for x
Now, we will isolate \( x \):

\[ 20x - 9x = 3 + 8 \]
\[ 11x = 11 \]
\[ x = 1 \]

### Conclusion
The value of \( x \) that satisfies the equation is:

\[ \boxed{1} \]