You need to take logarithms both sides such that:

`lg(x^(2+lgx)) = lg(100x)`

You need to convert the logarithm of the product to the right side, in the sum of logarithms such that:

`lg(x^(2+lgx)) = lg 100 + lg x`

You need to use power property of logarithms such that;

`(2+lg x)*lg x = lg(10^2) + lg x`

You need to open the brackets to the left side such that:

`2 lg x + lg^2 x = 2 + lg x`

You should come up with the substitution `lg x = y` such that:

`y^2 + y - 2 = 0`

You should use quadratic formula such that:

`y_(1,2) = (-1+-sqrt(1+8))/2`

`y_(1,2) = (-1+-sqrt9)/2`

`y_(1,2) = (-1+-3)/2`

`y_1 = 1 ; y_2 = -2`

You should solve for x the equations`lg x = 1` and `lg x = -2` such that:

`lg x = 1 =gt x = 10`

`lg x = -2 =gt x = 10^(-2) =gt x = 1/100`

**Hence, evaluating the solutions to the given equation yields `x = 10 ` and `x = 1/100.` **