Taylor approximation

first-order

$$f(x)\approx f(a)+f^{\prime }(a)(x-a)$$

so the approximation of $f(x)=\log (1+x)$ around $x=0$ is

$$\log (1+x)\approx x$$

second-order

$$f(x)\approx f(a)+f^{\prime }(a)(x-a)+{\displaystyle \frac{1}{2}}{f}^{″}(x)(x-a{)}^2$$

example

given $g(X)$ & $\mu =E(X)$

first-order

$$g(X)\approx g(\mu )+g^{\prime }(\mu )(X-\mu )$$

$$Var(g(X))=[g^{\prime }(\mu ){]}^{2}Var(X)$$

second-order

$$g(X)\approx g(\mu )+g^{\prime }(\mu )(X-\mu )+{\displaystyle \frac{1}{2}}{g}^{″}(X)(X-\mu {)}^2$$

$$E[g(X)]\approx g(\mu )+{\displaystyle \frac{1}{2}}{g}^{″}(\mu )Var(X)=g(\mu )+{\displaystyle \frac{1}{2}}{\displaystyle \frac{g^{″}(\mu )}{[g^{\prime }(\mu ){]}^2}}Var(g(X))$$

when $g(X)=\log (X)$

$$E[\log (X)]\approx \log (\mu )-{\displaystyle \frac{1}{2}}Var(\log (X))$$

$$\log E(X)=E(\log X)+{\displaystyle \frac{1}{2}}Var(\log X)$$

$$E(X)=e^{E(\log X)+\frac{1}{2}Var(\log X)}$$