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Is it the same as $\ln(x)^2$? And if so, is it equal to $2\ln(x)$?

Thanks in advance.

3SAT
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Healshine
  • 91

5 Answers5

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No it is not!

Note that $\log (x^2)=2\log(x)$ but $\log^2(x)\color{red}\neq 2\log(x)$

Generally $\log(x^n)=n\log(x)$

3SAT
  • 7,512
2

$\ln^2(x)=\ln(x)^2 \not = 2 \ln(x)$.

Perhaps you are thinking of the rule $\ln(x^k)=k\ln(x)$. That is only true when the exponent is inside the logarithm. Also, $\ln(x)$ is just like any 'ol number, so multiplying it with $2$ is not (in most cases) equivalent to setting it to the power of $2$.

Bobson Dugnutt
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2

Usually, $\ln^2x=(\ln x)^2$. Note that $\ln(x^2)=2\ln x$. So, $\ln^2x\ne 2\ln x$.

Tim Raczkowski
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2

$\ln^2 x$ probably means $(\ln x)^2$, in analogy with expressions like $\sin^2 x$ etc., so on a normal AOS calculator you would typically:

  • type in the value of $x$
  • press the "$\ln$" button
  • press the "$x^2$" button

Notation varies among calculators, of course, but this is the general idea.

MPW
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Unfortunately $\ln^2 x$ is ambiguous: sometimes it means $(\log x)^2$ and sometimes it means $\log\log x$.

In either case it's different from $\log(x^2)=2\log x$.

Michael Hardy
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  • When iteration is a possibility, I usually distinguish the iterate $f^n(x)$ from the power $f(x)^n$, both of which are distinct from the argument carrying the exponent in $f(x^n)$. However, I have never seen iteration notation used in the context of "named" function symbols like $\log$ or $\sin$, so I would always consider the $2$ in $\log^2 x$ to be an exponent. (Of course, in context, it is perfectly fine to use this to mean an iterate--I'm just saying I think that's pretty rare, or at least I haven't seen it.) +1. – MPW Feb 23 '16 at 14:10