Answer: (a)(i) \(x\gt \frac35\). (ii) \(x=\frac{13}{5}\). (b) \(y=\frac{\sqrt[4]{x}}{6}\).

Answer: (a)(i) \(x\gt \frac35\). (ii) \(x=\frac{13}{5}\). (b) \(y=\frac{\sqrt[4]{x}}{6}\).

(a)(i) The logarithm \(\lg(5x-3)\) exists when its argument is positive:

\(5x-3\gt 0\).

Hence

\(x\gt \frac35\).

(a)(ii) From

\(\lg(5x-3)=1\),

we have

\(5x-3=10\).

Therefore \(5x=13\), so

\(x=\frac{13}{5}\).

(b) Rewrite the constant 4 as a logarithm to base \(y\):

\(4=\log_y(y^4)\).

Also,

\(\frac12\log_y64=\log_y\sqrt{64}=\log_y8\).

So

\(\log_y x=\log_y(y^4)+\log_y8+\log_y162\).

Combine the logarithms:

\(\log_y x=\log_y(1296y^4)\).

Thus

\(x=1296y^4\).

Since \(1296=6^4\),

\(y^4=\frac{x}{6^4}\).

As \(y\gt 0\),

\(\displaystyle y=\frac{\sqrt[4]{x}}{6}\).