Shorthand notation for powers of logarithmic functions

Question

I've got an assignment here, and one question is throwing me off, as I've never seen it written like this before..

$$\int\frac{\ln^3 x}{x}\ dx$$

Is this the same as $$\int\frac{(\ln x)^3}{x}\ dx\;\;?$$

Answer 1

Unfortunately, "shorthand" can lead to ambiguity:

$\ln^3 x = (\ln (\ln(\ln x))\;?\quad$ or $\quad \ln^3x = (\ln x)^3\;?$

But as you suspected, in this context, and given the integral, I'm am quite sure that $\ln^3x = (\ln x)^3,\;$ much like $\;\sin^2(x) = (\sin x)^2.\;$ So your integral amounts to:

$$\int\frac{\ln^{3}x}{x}\ dx = \int \frac{(\ln x)^3}{x} \,dx$$

Let $u = \ln(x),;\;du = \dfrac{dx}{x}$

$$\int \frac{(\ln x)^3}{x} \,dx = \int u^3\,du = \frac14 u^4 + C $$ $$= \frac14 (\ln x)^4 + C = \frac 14\ln^4x + C$$

Answer 2

Unfortunately two conventions prevail: $\ln^3 x$ can mean $\ln\ln\ln x$ or it can mean $(\ln x)^3$. The first one is reasonable; the second one is unfortunately also seen. I would write $(\ln x)^3$, or $(\log x)^3$, or $(\log_e x)^3$, if I meant $(\log_e x)^3$.

It became universally standard to construe $\sin^3 x$ to mean $(\sin x)^3$, over the objection of Carl Gauss (the most famous person to live on earth in the 19th century, except perhaps those who did not work in the physical and mathematical sciences), who said $\sin^3 x$ ought to mean $\sin\sin\sin x$. You can't push back the tide with a pitchfork.