= Solution
Work first on $x>0$, where the proposed <logarithm> is defined. Substitution of $y=x^m$ into the <Cauchy-Euler differential equation> gives the <indicial equation>
$$
m(m-1)+m+1=m^2+1=0.
$$
Its distinct <indicial roots> are $m=\pm i$. Using $x^{\pm i}=e^{\pm i\log x}$ and <Euler's formula>, a real <basis> of solutions is $\cos(\log x),\sin(\log x)$. Their <Wronskian> is $1/x$, so they are <linearly independent> and span the two-dimensional <solution space of a homogeneous linear differential equation>. Therefore
$$
\boxed{y(x)=A\cos(\log x)+B\sin(\log x),\qquad x>0.}
$$
For the requested <change of variables>, set $Y(z)=y(e^z)$. The <chain rule> gives
$$
y'=\frac{Y'}x,\qquad y''=\frac{Y''-Y'}{x^2}.
$$
The transformed <second-order linear differential equation> is $Y''+Y=0$. Its <characteristic roots> are again $\pm i$, giving $Y(z)=A\cos z+B\sin z$. Substituting $z=\log x$ recovers precisely the previous <general solution>.
If solutions on $x<0$ are wanted, the same calculation uses $z=\log|x|$ and gives independent constants on that interval. Zero is a <singular point of a differential equation>; the logarithmic oscillations do not define a nonzero continuous solution through it.
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