# How is the utility function with constant relative risk-aversion obtained?

In this slide, it says that constant relative risk-Aversion utility function have this form.

$$u(x) = \frac{1}{1-b} x^{1-b}$$ for $$b≠1$$

$$u(x) = In(x)$$ for $$b=1$$

When I tried to derive the utility function from $$b = -x \frac{u''(x)}{u'(x)}$$ (the representation of CRRA), I got $$u(x) = x^{1-b}$$.

While I understand that preference will not be changed by linear transformation of the utility function, I wonder why is it necessary to make the utility function be $$u(x) = \frac{1}{1-b} x^{1-b}$$ for $$b≠1$$.

Also how is the "$$u(x) = In(x)$$ for $$b=1$$" obtained?

In the slide, we're given the marginal utility (or the derivative of the utility function) as $$m(x) = x^{-b}$$. The utility function whose derivative is $$m(x)$$ is
$$\begin{eqnarray*} u(x) = \int m(x) dx = \int x^{-b} dx = \begin{cases} \frac{1}{1- b} x^{1-b} & \text{when } b \neq 1 \\ \ln x & \text{when } b = 1 \end{cases} \end{eqnarray*}$$
Verify that $$u'(x) = m(x)$$.