Answer
b) $20$
Work Step by Step
a) Let's note:
$$\log_bx=m, \log_a x=n$$
We have:
$$x=b^m, x=a^n$$
$$b^m=a^n\Rightarrow b=a^{n/m}$$
Let's calculate:
$$\frac{\log_a x}{\log_a b}=\frac{n}{\log_a a^{n/m}}=\frac{n}{\frac{n}{m}}=m$$
But $m=\log_bx$, so we got
$$\log_bx=\frac{\log_a x}{\log_a b}$$
b) We have:
$$\begin{aligned}
\log_2 81\cdot\log_3 32&=\log_2 (3^4)\cdot\log_3 (2^5)\\
&=(4\log_2 3)(5\log_3 2)\\
&=20\log_2 3\cdot\log_3 2.
\end{aligned}$$
We use the change of base formula with for $\log_3 2$ with $b=3$ and $x=2$:
$$20\log_2 3\cdot\log_3 2=20\log_2 3\cdot\frac{\log_2 2}{\log_2 3}=20\cdot 1=20$$
