Solving equation involving a variable and its logarithm

[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.

 

[ShayanJ]

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

 

[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.

 

[Erland]

There is no analytical solution.

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

 

[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.

 

[HallsofIvy]

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

 

[Borek]

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