<s>[INST] Let $x>1,y>1$ and $\left(\log_e x \right)^2+\left(\log_e y \right)^2=\log(x^2)+log(y^2)$, then find the maximum value of $x^{\log_e y}$ Let $x>1,y>1$ and $\left(\log_e x \right)^2+\left(\log_e y \right)^2=\log(x^2)+log(y^2)$, then find the maximum value of $x^{\log_e y}$
My attempt is as follows:
As $x>1,y>1$ , so $\log_e x>0, \log_e y>0$, hence we can apply $AM>=GM$
$$\dfrac{log_e x+log_e y}{2}>=\sqrt{\log_e x\log_e y}$$
As both the sides are positive, so we can square both the sides without breaking the inequality.
$$(\log_e x)^2+(\log_e y)^2+2\log_e x\log_e y>=4\log_e x\log_e y$$
Using the given condition $\left(\log_e x \right)^2+\left(\log_e y \right)^2=\log(x^2)+log(y^2)$
$$(\log_e x^2)+(\log_e y^2)>=2\log_e x\log_e y$$
$$(\log_e x)+(\log_e y)>=\log_e x\log_e y$$
$$\log_e xy>=\log_e x^{\log_e y}$$
As $e>1$, so we can safely write $xy>=x^{\log_e y}$
But actual answer is $e^4$, I am not able to think of any other way. Please help me in this.
 [/INST] Here is a way where you do not need any standard inequality like AM-GM or Cauchy-Schwarz. 
The fist step is, as already shown in an other solution, to get rid of the logarithms by substituting $x=e^a$, $y=e^b$ with $a,b >0$ which gives:


*

*$a^2+b^2 = 2(a+b)$ and

*maximize $x^{\ln y} =e^{\ln x\cdot \ln y} = e^{ab}$.


Now, observe that by using square completion you get
$$a^2+b^2 = 2(a+b) \Leftrightarrow a^2-2a + b^2+2b = 0 \Leftrightarrow \boxed{(a-1)^2+(b-1)^2=2}$$
This is a circle around $(1,1)$ with radius $\sqrt{2}$. So, you can write


*

*$a-1 =\sqrt{2}\sqrt{\cos t} \Leftrightarrow \boxed{a = 1+ \sqrt{2}\cos t}$

*$b-1 =\sqrt{2}\sqrt{\sin t} \Leftrightarrow \boxed{b = 1+ \sqrt{2}\sin t}$
Now, lets maximize $ab$ and let's check at the end whether the maximum is attained for a $t \in [0,2\pi]$ such that $a,b >0$. Here you may use


*

*$(1)$: $\sin\frac{\pi}{4} =\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$

*$(2)$: $(1)\Rightarrow \frac{\sqrt{2}}{2}(\cos t + \sin t) = \sin \left(t+\frac{\pi}{4}\right)$

*$(3)$: $2\sin t \cos t = \sin 2t$
\begin{eqnarray*}ab
& = & (1+ \sqrt{2}\cos t)(1+ \sqrt{2}\sin t)\\
& = & 1+ \sqrt{2}(\cos t + \sin t) + 2\cos t \sin t \\
& \stackrel{(2),(3)}{=} & 1 + 2\sin \left(t+\frac{\pi}{4}\right) + \sin 2t \\
& \leq & 1+2+1 = \boxed{4}
\end{eqnarray*}
Since the maximum is indeed attained for $\boxed{t=\frac{\pi}{4}}$ which satisfies the condition $a,b>0$, you are done.
So, the maximum value of $x^{\log_e y}$ under the given constraints is $\boxed{e^4}$.
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