Solve the following equations : `(i) log_(x)(4x-3)=2` `(ii) log_2)(x-1)+log_(2)(x-3)=3`
`(iii) log_(2)(log_(8)(x^(2)-1))=0` `(iv) 4^(log_(2)x)-2x-3=0`

Answer

Let's solve the equations step by step.

### (i) Solve the equation: \( \log_{x}(4x-3) = 2 \)

1. **Rewrite in exponential form**:
\[ 4x - 3 = x^2 \]

2. **Rearrange the equation**:
\[ x^2 - 4x + 3 = 0 \]

3. **Factor the quadratic**:
\[ (x - 3)(x - 1) = 0 \]

4. **Find the roots**:
\[ x = 3 \quad \text{or} \quad x = 1 \]

5. **Check the validity of the solutions**:
- For \( x = 1 \): \( 4(1) - 3 = 1 \) and \( \log_{1}(1) \) is undefined.
- For \( x = 3 \): \( 4(3) - 3 = 9 \) and \( \log_{3}(9) = 2 \) is valid.

**Final answer**: \( x = 3 \)