Find x, if :
(i) `log_(x) 625 = -4`
(ii) `log_(x) (5x - 6) = 2`.

Answer

Let's solve the given logarithmic equations step by step.

### Part (i): Solve `log_(x) 625 = -4`

1. **Rewrite the logarithmic equation in exponential form**:
\[ x^{-4} = 625 \]
This is derived from the definition of logarithms, which states that if \( \log_b(a) = c \), then \( b^c = a \).

2. **Rewrite 625 as a power of 5**:
\[ 625 = 5^4 \]
Therefore, we can substitute this into our equation:
\[ x^{-4} = 5^4 \]

3. **Set the bases equal**:
Since the bases are equal, we can equate the exponents:
\[ -4 = 4 \log_{5}(x) \]

4. **Solve for \( x \)**:
Rearranging gives:
\[ \log_{5}(x) = -1 \]
Now, converting back to exponential form:
\[ x = 5^{-1} = \frac{1}{5} \]

Thus, the solution for part (i) is:
\[ \boxed{\frac{1}{5}} \]