Log^2(x) and log(x)^2 are the same or not?

[ddr]

I hope to not seem to much silly but:

log^2(x)

and

log(x)^2

are the same or not?

thanx

 

[AlephZero]

The usual notation for "log x times log x" is log^2 x (similarly to sin^2 x, means "sin x times sin x").

The usual notation for " the log of (x times x)" is log x^2.

In your examples the () are not really needed unless "x" is a more complicated expression, like log^2(x+y) or log (x+y)^2.

BTW it's not a silly question - if you don't know what any mathematical notation means, you need to find out, otherwise you can't use it properly!

 

[neutrino]

What is log^2(x)?

Did you mean this: $$\log_2(x)$$? In that case they may not.

$$\log_2(x)$$ is the logarithm of x to the base 2.

$$(\log(x))^2$$ is the square of the logarithm of x to some base, usually taken to be 10.

$$(\log(x^2))$$ is the logarithm of x^2. Again, the base may be 10.

 

[neutrino]

The usual notation for "log x times log x" is log^2 x (similarly to sin^2 x, means "sin x times sin x").

I've never come across that notation before!

 

[HallsofIvy]

I'm with neutrino: I've never seen log(x)². Either log²(x) or (log(x))² means (log(x))(log(x)). log(x²) means, of course log((x)(x)) (which is equal to 2log(x)). Of course, since people do not always use parentheses with logarithm (or trig functions), you might see log x² which I would interpret as log(x²)= 2log(x).

 

[AlephZero]

Well, at least nobody's suggested that log^2 x meant log(log x) - and it doesn't, mean that, of course.

I can't think of a reason why you would ever want to write (log x)^2 except in a calculus exercise like "integrate the function (log x)/x".

But if I did see it, I would have assumed it meant the same as sin^2 x etc, unless the context implied something different.

 

[neutrino]

I'm with neutrino: I've never seen log(x)². Either log²(x) or (log(x))² means (log(x))(log(x)).

Err...it's the log²(x) that I haven't come across. It has always been (log(x))² for me.