Solving equation involving a variable and its logarithm

1. Aug 15, 2014

JulieK

Can you suggest a general analytical solution to the following equation

$\ln(x^{3/2})-bx-c=0$

where $x$ is real positive variable and $b$ and $c$ are real positive
constants.

2. Aug 15, 2014

ShayanJ

There is no analytical solution. It should be solved numerically.

3. Aug 15, 2014

Simon Bridge

Sure - it is the intersection of $y=\ln(x)$ with the line $y=2(bx+c)/3$.
Note: $\ln(x)$ is only defined for $x>0$.

Solutions will not exist for all a and b.
In general you'll need a numerical solution.

4. Aug 15, 2014

Erland

You're probably right, but ... can you prove it?

5. Aug 15, 2014

da_nang

Depends on whether or not you classify the Lambert W function as an analytical solution (I doubt the mathematics community does).

$$\ln(x^{\frac{3}{2}}) = b x + c \\ x = \exp( \frac{2}{3}bx + \frac{2}{3}c) \\ x \exp(- \frac{2}{3} bx) = \exp( \frac{2}{3}c) \\ - \frac{2}{3} b x \exp(- \frac{2}{3}bx) = -\frac{2}{3} b \exp( \frac{2}{3}c) \\ - \frac{2}{3} b x = W_n(-\frac{2}{3} b \exp( \frac{2}{3}c)) \\ x = - \frac{3 W_n(-\frac{2}{3} b \exp( \frac{2}{3}c))}{2 b}$$

For a real $x > 0$, $b \neq 0$ and $n \in \mathbb{Z}$. Only $n=-1, 0$ can provide real solutions, though.

Last edited: Aug 15, 2014
6. Aug 15, 2014

HallsofIvy

Staff Emeritus
The W-function is certainly an analytic function so I would call that an "analytic solution".

7. Aug 15, 2014

Staff: Mentor

Looks like confusion between "analytic function" and "analytic expression".