I was trying to solve a problem. Due to solve, I tried to use change a variable but I saw, I got a conflict after do it.
Let's assume $y=1+n+n^2$ and lets try to change n variable with $n^k=\log(m^k)$
$y=\log(1\cdot m^0)+\log(m)+\log(m^2)=0+3⋅\log(m)$
If we try to go back, $\log(m)=n$
$y=3⋅n$
And as you see, the original equation $y=1+n+n^2$ is not equal to $y=3⋅n$ If we calculate two equation for $2$, equation one is $7$ equation two is $6$.
What is the problem ? We always do it for integral or some other problems.
Thanks.
$$n^k=\log(m^k)$$ cannot hold for several values of $k$ simultaneously.
For instance,
$$n=\log(m)$$ does not imply
$$n^2=\log(m^2)$$
because that would mean $$n^2=\log(m^2)=2\log(m)=2n.$$