Let $\mathbf{x}$ be a uniform random variable over $[0,1]$ and $\mathbf{y}$ the random variable defined by $\mathbf{y}=e^{\mathbf{x}}$. I'd like to show that $$ E(\mathbf{y}|\mathbf{x}=x)=e^x $$ By definition, $$ \begin{align} E(\mathbf{y}|\mathbf{x}=x) &=\int_{-\infty}^{\infty}y f_{\mathbf{y}|\mathbf{x}}(x,y) dy \\ &=\int_{-\infty}^{\infty}y \frac{f_{\mathbf{x},\mathbf{y}}(x,y)}{f_{\mathbf{x}}(x)} dy \end{align} $$ Thus, I first tried to find the joint p.d.f., $$f_{\mathbf{x},\mathbf{y}}(x,y)$$ but it became $0$ and I cannot obtain the conditional expectation.
My attempt
First, I computed the c.d.f. of $\mathbf{y}$ by change of variable: $$ \begin{align} F_{\mathbf{y}}(y) &= P(\mathbf{y}\leq y) \\ &=P(e^\mathbf{x}\leq y)\\ &=P(\mathbf{x}\leq \log(y)) \end{align} $$ Therefore, when $0\leq \log(y)\leq 1$, or $1\leq y \leq e$, the c.d.f. is $$ \begin{align} F_{\mathbf{y}}(y) &=\int_{0}^{\log(y)}f_{\mathbf{x}}(x)dx \\ &= \int_{0}^{\log(y)}1dx \\ &= \log(y) \\ \therefore f_{\mathbf{y}}(y) &= \frac{d}{dy}F_{\mathbf{y}}(y) = \frac{1}{y}\\ \therefore f_{\mathbf{y}}(y) &= \begin{cases} \frac{1}{y} & 1\leq y \leq e\\ 0 & \text{elsewhere} \end{cases} \end{align} $$ Similarly, we consider the joint c.d.f. of $\mathbf{x}$ and $\mathbf{y}$: $$ \begin{align} F_{\mathbf{x},\mathbf{y}}(x,y) &= P(\mathbf{x}\leq x, \mathbf{y}\leq y)\\ &= P(\mathbf{x}\leq x, e^\mathbf{x}\leq y)\\ &= P(\mathbf{x}\leq x, \mathbf{x}\leq \log(y))\\ &= \begin{cases} P(\mathbf{x}\leq x) & \text{if }\ x\leq \log(y) \\ 0 & \text{elsewhere} \end{cases} \\ \\ P(\mathbf{x}\leq x\leq \log(y)) &= \int_{0}^{\log{x}}f_{\mathbf{x}}(x)dx\\ &= \log(x) \\ \\ \therefore \ F_{\mathbf{x},\mathbf{y}}(x,y) &= \begin{cases} \log(x) & \text{if }\ x\leq \log(y) \\ 0 & \text{elsewhere} \end{cases} \end{align} $$ Finally, we compute the joint p.d.f. by a partial derivative of the joint c.d.f.: $$ \begin{align} f_{\mathbf{x},\mathbf{y}}(x,y) &= \frac{\partial^2}{\partial x \partial y}F_{\mathbf{x},\mathbf{y}}(x,y)\\ &= 0 \end{align} $$ Strangely, my result becomes $0$. I'm sure I'm messing up something in my calculation. I appreciate any suggestions or corrections!