0

$$\left(\frac{\ln{x}}{x}\right)^3 = \frac{(\ln x)^3}{x^3} $$

Is $(\ln x)^3 $ same as $\ln x^3$ ?

Meaning when before I differentiate it, I can move the $3$

$$ 3 \ln x. $$

Xander Henderson
  • 29,772
user307640
  • 2,632
  • 2
  • 17
  • 27
  • The identity $(\ln x)^3=3\ln x$ holds for literally three real numbers. –  Aug 18 '17 at 13:58

2 Answers2

3

No. Generally speaking, function composition is not commutative.

$$(f\circ g)(x)\ne(g\circ f)(x)$$ or

$$f(g(x))\ne g(f(x)).$$

$$(\ln x^3)'=(3\ln x)'=\frac3x\ne(\ln^3x)'=3\ln^2x(\ln x)'=\frac{3\ln^2 x}x.$$

2

No, $$\ln(x^3)\neq(\ln(x))^3$$ in general. You would need to use the chain rule to differentiate $(\ln(x))^3$.

Dave
  • 13,568
  • 5
    Also, $(\ln(x))^3$ is commonly denoted as $\ln^3x$. – Puding Aug 18 '17 at 13:42
  • @Puding It is? I've only seen that convention used commonly for the trigonometric functions, not for other functions. – Deepak Aug 18 '17 at 13:44
  • @Deepak It is a very common notation, at least in theoretical computer science. Parentheses are a pain to type and parse, $(\log n)^2$ is much less convenient than $\log^2 n$. – Clement C. Aug 18 '17 at 13:47
  • Yes, it is mainly used with trigonometric functions, because they are much more often raised to powers than logarithm, but there is no reason why the same notation couldn't be applied to any function. See this question. @Deepak – Puding Aug 18 '17 at 13:49