Logarithmic question: $n^{1/\lg n}=2$

Question

How is:

$$n^{\Large\frac 1{\lg n}} = 2\ ? $$

I don't understand this is their any formula to calculate this

what is the difference between $\lg n$ & $\log n$? Is logarithm base $2$ ?

Answer 1

I think here $\lg$ means $\lg 2$, so $n^{\frac{1}{\lg_2 n}}=n^{\lg_n 2}=2$.

Answer 2

In your question, $\lg$ is base 2. By applying $\lg$ to your equation, you get $\lg( n^{1/\lg n} ) = \lg 2.$ Using the law $b \lg a = \lg(a^b)$, we find $1 = \lg 2$ which is true, since $\lg$ has base 2.