Name for a Logarithm Identity/Property

I come across the same property and struggled to grasp the intuition until I try different numbers for n and x for in your equation.

$$n^{\log_bx} = x^{\log_bn}$$

Let's think about n=2, x=8 and assume b=10 and then try to generalize the concept.

$$2^{\log_{10}8} = 8^{\log_{10}2}$$ $$2^{\log_{10}2^3} = 2^{3\log_{10}2}$$ $$2^{3\log_{10}2} = 2^{3\log_{10}2}$$

So we can start to think with $n=x^k$ and it always results as following while log base doesn't matter except being the same:

$$n^{\log_bx} = x^{\log_bn} $$ $$x^{k\log_bx} = x^{\log_bx^k} $$ $$ x^{k\log_bx} = x^{k\log_bx} $$