(i) If log(a + 1) = log(4a - 3) - log 3, find a.
(ii) If 2 log y - log x - 3 = 0, express x in termss of y.
(iii) Prove that : `log_(10) 125 = 3(1 - log_(10)2)`.

Answer

Let's solve the given questions step by step.

### Part (i)
Given:
\[ \log(a + 1) = \log(4a - 3) - \log 3 \]

**Step 1:** Apply the property of logarithms that states \( \log A - \log B = \log \left(\frac{A}{B}\right) \).
\[ \log(a + 1) = \log\left(\frac{4a - 3}{3}\right) \]

**Step 2:** Since the logarithms are equal, we can equate the arguments.
\[ a + 1 = \frac{4a - 3}{3} \]

**Step 3:** Multiply both sides by 3 to eliminate the fraction.
\[ 3(a + 1) = 4a - 3 \]
\[ 3a + 3 = 4a - 3 \]

**Step 4:** Rearrange the equation to isolate \(a\).
\[ 3 + 3 = 4a - 3a \]
\[ 6 = a \]

**Final Answer for Part (i):**
\[ a = 6 \]