$x^y=z $
Proof that
$x^n/z=y$
I was calculating the cube root of $2$ by hand and when checking it out, I noticed its square is close to the value of logarithm of $3$ in base $2$. A little tweaking and I got the exact value for $n$ when $z$ is set at $3$. I wonder if there is detailed proof of such operations. Thanks.
On
Adding to abc..., note that: $$\log_b{x}=y \Rightarrow b^y=x$$
You're saying: $$2^{\frac{2}{3}}\approx\log_2{3}$$
If we break this into the components of a logarithm, we have: $b=2$, $y=\frac{2}{3}$, and $x=3$.
That would imply that $2^{\frac{2}{3}}=3$ by our definition of a logarithm.
Clearly, this is a contradiction though as $2^{\frac{2}{3}}\approx1.5874$. Therefore, this is just an interesting coincidence, not consequence of any "rule".
To me it's a coincidence.
$\log_2 3-2^{\frac{2}{3}}=0.002438551247043293297966695324491751631678920207371949352$