## anonymous 5 years ago Use a graphing utility to solve the equation. ln x=(x^3)-3

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.