How find this minimum of $\sqrt{(x-a)^2+(3-x-\lg{(a)})^2}+\sqrt{(x-b)^2+(3-x-10^{b})^2}$ let $x,a,b\in R$,and $a>0$,find this follow minimum of the value 
$$\sqrt{(x-a)^2+(3-x-\lg{(a)})^2}+\sqrt{(x-b)^2+(3-x-10^{b})^2}$$
I see this two function
$$f(x)=10^a,g(x)=\lg{(x)}$$ are Mutually inverse function
maybe can use
$$\sqrt{x^2+y^2}+\sqrt{a^2+b^2}\ge\sqrt{(a+x)^2+(b+y)^2}$$
But I can't,Thank you

HINT :
Let $A(a,\log_{10} (a)), B(b,10^{b})$. Also, let $L$ be the line $y=3-x$.
What you want is the minumum of
$$|AP|+|BP|$$
where $P (x,y)$ is a point on $L$ and $|AP|$ represents the distance between $A$ and $P$.
You can solve your question geometrically.
