$1^x = 1^y$ and $x,y$ belongs to Real Numbers.

Question

$1^x = 1^y$, and $x,y \in \mathbb{R}$.

Following the rule, same base has powers equal every $x$ should be equal to every $y$. $$1^x = 1^y$$ $$x = y$$ What went wrong?

Answer 1

$1^x=1^y$ does not imply $x=y$:

$$x\cdot \ln(1)=y\cdot\ln(1)$$ $$x \cdot 0=y\cdot 0$$

Note that $x,y$ can be anything- including different values, i.e., $x ≠y.$

The rule you mention is incorrect.

Answer 2

Can you conclude that $2=3$ from this equation $ 1^2=1^3$?

Answer 3

The function $x\to 1^x$ isn't injective.