Let f(x) = [ln(x)]^5; I am assuming you mean this? Or, do you mean ln(x^5)?

I'll do both; I'll assume for case 1 you mean the first and for case 2 you mean the latter:

Case 1:

f(x) = [ln(x)]^5

f'(x) = 5*[ln(x)]^4*(1/x) = [5*[ln(x)]^4]/x

f''(x) = Use quotient rule:

Quotient rule states:

[f(x)/g(x)]' = [g(x)*f'(x) - f(x)*g'(x)]/[g(x)]^2

[x*20*[ln(x)]^3*(1/x) - [5*[ln(x)]^4]*1]/[x]^2

= [20*[ln(x)]^3 - 5*[ln(x)]^4]/x^2

Case 2:

f(x) = ln(x^5)

f'(x) = 5/x

f''(x) = -5/(x^2)

I think the above is pretty easy to see, so I think you mean the first one.