Answer: (a) \(\log_2\left(\frac{8a^4}{b}\right)\). (b) \(x=100\) or \(x=\frac{1}{100}\).

Apply the laws of logarithms carefully, paying attention to the base of each logarithm and any restrictions on the argument.

For part (a), write \(3=\log_2 8\), and use the logarithm laws:

\(3+4\log_2a-\log_2b=\log_28+\log_2a^4-\log_2b.\)

Therefore

\(3+4\log_2a-\log_2b=\log_2\left(\frac{8a^4}{b}\right).\)

For part (b), use change of base:

\(\log_x10=\frac{1}{\lg x}.\)

The equation \(\lg x=4\log_x10\) becomes

\(\lg x=\frac{4}{\lg x}.\)

So \((\lg x)^2=4\), and hence \(\lg x=2\) or \(\lg x=-2\).

Thus \(x=100\) or \(x=10^{-2}=\frac{1}{100}\).

This gives the required answer: (a) \(\log_2\left(\frac{8a^4}{b}\right)\). (b) \(x=100\) or \(x=\frac{1}{100}\).