Is $g(x)=\log x$ convex function?

Question

The graph of convex function is :

In a book it is written that $g(x)=\log x$ is strictly convex function.

So i searched for graph of $g(x)=\log x$ and found that

Though it has been said that $g(x)=\log x$ is strictly convex function, comparing these two graph it seems to me $g(x)=\log x$ is concave function .

Where am i doing mistake ?

Answer 1

The function $g(x)$ is a concave. You can see from your graph that the line passing through two given points on the curve lies below the graph of $g$, not above the graph (which you would get with a convex function).

Answer 2

The function is concave if it's second derivative is negative. We have that $$ g''(x)=(\log(x))''=\biggl(\frac1x\biggr)'=-\frac1{x^2} $$ for $x>0$. Hence, $g(x)$ is a concave function.