- #1
ddr
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I hope to not seem to much silly but:
log^2(x)
and
log(x)^2
are the same or not?
thanx
log^2(x)
and
log(x)^2
are the same or not?
thanx
Log^2(x) and log(x)^2 are the same or not?
- Thread starter ddr
- Start date
In summary, there are various notations for expressing "log x times log x" and "log of (x times x)". The most common ones are log^2 x and (log x)^2, respectively. The notation log2(x) is not commonly used. It is important to understand the notation in order to use it correctly.
- #2
AlephZero
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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!
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!
- #3
neutrino
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What is log^2(x)?
Did you mean this: [tex]\log_2(x)[/tex]? In that case they may not.
[tex]\log_2(x)[/tex] is the logarithm of x to the base 2.
[tex](\log(x))^2[/tex] is the square of the logarithm of x to some base, usually taken to be 10.
[tex](\log(x^2))[/tex] is the logarithm of x^2. Again, the base may be 10.
Did you mean this: [tex]\log_2(x)[/tex]? In that case they may not.
[tex]\log_2(x)[/tex] is the logarithm of x to the base 2.
[tex](\log(x))^2[/tex] is the square of the logarithm of x to some base, usually taken to be 10.
[tex](\log(x^2))[/tex] is the logarithm of x^2. Again, the base may be 10.
- #4
neutrino
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AlephZero said:
I've never come across that notation before!
- #5
HallsofIvy
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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).
- #6
AlephZero
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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.
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.
- #7
neutrino
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HallsofIvy said:
Err...it's the log2(x) that I haven't come across.
It has always been (log(x))2 for me.FAQ: Log^2(x) and log(x)^2 are the same or not?
Are log^2(x) and log(x)^2 the same function?
No, log^2(x) and log(x)^2 are not the same function. While they may have similar notation, they represent two different mathematical concepts. Log^2(x) indicates taking the logarithm of x and then squaring the result, while log(x)^2 indicates taking the logarithm of x and then raising it to the power of 2.
Can log^2(x) and log(x)^2 be used interchangeably?
No, log^2(x) and log(x)^2 cannot be used interchangeably. They have different mathematical properties and will produce different results when applied to the same input. It is important to use the correct notation for the intended operation.
What is the difference between log^2(x) and log(x)^2?
The main difference between log^2(x) and log(x)^2 is the order of operations. In log^2(x), the logarithm is taken first and then the result is squared. In log(x)^2, the logarithm is taken and then the result is raised to the power of 2. This results in different values for the same input.
Can log^2(x) or log(x)^2 be simplified?
Yes, both log^2(x) and log(x)^2 can be simplified using logarithm rules. For log^2(x), it can be simplified to 2log(x). For log(x)^2, it can be simplified to log(x^2). However, these simplifications are only valid if the base of the logarithm is not specified.
Do log^2(x) and log(x)^2 have the same domain and range?
Yes, log^2(x) and log(x)^2 have the same domain and range. Both functions are defined for all positive real numbers, with the domain being (0, ∞) and the range being (-∞, ∞). However, they are still different functions with different outputs for the same input.