# How do you solve #ln(x) = x^3 - 3#?

##### 1 Answer

#### Explanation:

We have:

# lnx=x^3-3 #

This equation cannot be solved analytically, so first we graph the functions to get a "feel" for the solutions:

So, we establish that there are two solutions, approximately

# x = g(x) # and use an iteration#x_(n+1) = g(x_n) #

There will many functions,

**Root 1:** For

We could try:

# lnx = x^3-3 => x^3=3+lnx#

# :. x = root(3)(3+lnx) #

So we will try the iterative equation:

# x_0 \ \ \ \ = 1.5 #

# x_(n+1) = root(3)(3+lnx_n)#

Using excel we can quickly process the iterative equation to any degree of accuracy. Here we work to 5dp:

Put

As it happens, and attempt of using

**Root 2:** For

# lnx = x^3-3 => x = e^(x^3-3) #

So we will try the iterative equation:

# x_0 \ \ \ \ = 0.5 #

# x_(n+1) = e^((x_n)^3-3) #