Question

Use a graphing utility to solve the equation.

ln x=(x^3)-3

Answer 1

Mathematica 8 was used for this response. "ln" is spelled "Log" below.
Refer to the attachment, a plot of x^3-3-Log[x] from x=0 to x=2.

To the casual observer the roots are approximately 0.05 and 1.5 .
More exact values for the two root are shown below:

FindRoot[x^3 - 3 - Log[x], {x, .01}] = .0497932...
FindRoot[x^3 - 3 - Log[x], {x, 1.4}] = 1.50499...

The symbolic solutions are:
\[-(-1)^{2/3} \left(\frac{1}{3} \text{ProductLog}\left[-\frac{3}{e^9}\right]\right)^{1/3} \]
\[-(-1)^{2/3} \left(\frac{1}{3} \text{ProductLog}\left[-1,-\frac{3}{e^9}\right]\right)^{1/3} \]
From the net:
ProductLog, or W cannot be expressed in terms of more common elementary functions.