To solve this equation, we can start by simplifying the left side:

lg(10x) lg(0.1x) = lg(10x 0.1x) = lg(x)

Now, we can simplify the right side:

lg(x^3) - 3 = 3lg(x) - 3

Now, we can set the left and right sides equal to each other:

3lg(x) - 3 = lg(x)

Subtract lg(x) from both sides:

2lg(x) - 3 = 0

Add 3 to both sides:

2lg(x) = 3

Divide by 2:

lg(x) = 3/2

Now, we can rewrite this in exponential form:

x = 10^(3/2)

x = 1000^0.5

x = 1000^1/2

x = √1000

x = 10√10

Therefore, the solution to the equation lg(10x) * lg(0.1x) = lg(x^3) - 3 is x = 10√10.