Separation of variables Calculus The given differential equation I need to solve is $dy/dx=1/x$ with the initial conditon of $x=1$ and $y=10$
My attempt: 
$dy=\frac 1x dx$
Integrating yields 
\begin{align*} 
   y&=\log x+C\\
   x=1 & y=10\\
 10 &=\log 1 + C\\
  10=C\\
\leadsto y&=\log x+10
\end{align*}
Something seems off with my work, if anyone could give me a clue where I went wrong. Thanks

Why do you think you are wrong? The derivative of $\ln(x)$ is in fact $1/x$. You are also satisfying the initial condition $y(1)=10$.
Also there is no need to use separation of variables. We can see directly that $$\frac{dy}{dx} = y'(x) = \frac{1}{x} \implies y(x) = \ln(x) + C.$$
Just as with elementary calculus and algebra, if you are not sure about your answer, then take a derivative and plug it back into the original equation.
