# Solve for positive $x$

Let $x>0$ be real. Find all solutions of $x^{0.3} = \ln x$ or more general equation $x^a= \ln x$ , where $0 < a < 1$.

• By taking $\ln$, this equation becomes $0.3 \ln x = \ln(\ln x)$ : it seems that it's impossible to find an analytic solution. – paf Aug 9 '16 at 17:48
• I don't know any analytical solutions. If numerical ones are ok, then you are looking for fixed points of the function $f(x) = \sqrt[\alpha]{\ln x}$ (or of a suitable local inverse, if that's more convenient). – avs Aug 9 '16 at 17:49
• I don't think it is easy to express in terms of elementary functions. if $a=1$ you could use the W function. – mathreadler Aug 9 '16 at 17:51
• WolframAlpha gives two solutions with Productlog function and its analytic continuation. One is $x = e^{\frac{-10}{3}W(-.3)}$ and $x = e^{-\frac{10}{3}W_{1}(-.3)}$, where $W(z)$ is the product log function and $W_k(z)$ is the analytic continuation of product log function. – Mathsira Aug 9 '16 at 17:51

Lets solve it:

$$x^a = \ln x \\ e^{x^a} = x \\ x = e^{x^a} \\ x^a = e^{a x^a} \\ a x^a = a e^{a x^a} \\ a x^a e^{-a x^a} = a \\ -a x^a e^{-a x^a} = -a$$

Let $s = -a x^a$. We then have the equation: $$s e^s = -a$$

Thus we can solve for $s$ if we say: $s = W(a)$, where $W$ is the inverse function of $f(s) = s e^s$, that is, $W = f^{-1}$, also called, Lambert-W Function. Be careful, there are two branchs in the real domain.

Therefore, we can now continue: $$-a x^a = W\left(-a\right)$$

Thus: $$x^a = -\frac{1}{a} W\left(-a\right)$$

Solutions of $x^a = \ln(x)$ are $(-W(-a)/a)^{1/a}$ where $W$ is a branch of the Lambert W function. In the case $a= 3/10$, there are two real solutions (corresponding to the principal and $-1$ branches of Lambert W), approximately $5.110722365$ and $379.0962301$.

EDIT: There are two real solutions if $0 < a < 1/e$, one if $a < 0$, and none if $a > 1/e$.

• Thank you. Could you please write the solution with more explanations ? – Mathsira Aug 9 '16 at 18:01
• Physicist137 explained it well. – Robert Israel Aug 9 '16 at 18:24