Name for a Logarithm Identity/Property

Question

I came across a neat logarithm fact today:

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

One simple proof is:

$\large \log_bx\cdot \log_bn=\log_bx\cdot \log_bn$

$\large \Rightarrow \log_bx^{log_bn}=log_bn^{log_bx}$

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

So you can swap the base with part of the exponent. Does this property have a name? What is the intuition? The other log properties are very intuitive to me, but not this one. Is it more easily derived from the usual four logarithm properties?

Answer 1

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:

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

Answer 2

$$ n^{\log_b x}=b^{\log_bn\cdot\log_bx}=\dots $$