Question

Given $3(\log 5-\log 3)-(\log 5-2\log 6)=2-\log n$, find $n$.

Answer 1

Given, $3(\log 5-\log 3)(\log 5-2\log 6)=2-\log n$

Simplify the given expression to find n,

$$\begin{gathered} \Rightarrow 3\log 5-3\log 3-\log 5+2\log (2\times 3)=2-\log n \\ \Rightarrow 2\log 5-3\log 3+2\log 2+2\log 3=2-\log n \\ \Rightarrow 2\log 5-\log 3+2\log 2=2-\log n \\ \Rightarrow \log 5^2-\log 3+\log 2^2+\log n=2 \\ \Rightarrow \log \left(25\times 4\times \frac{n}{3}\right)=2 \\ \Rightarrow \log \left(\frac{100n}{3}\right)=2 \\ \Rightarrow \left(\frac{100n}{3}\right)={10}^2 \\ \Rightarrow \left(\frac{100n}{3}\right)=100 \\ \Rightarrow n=3 \end{gathered}$$

Hence, $n=3$.