# Why is log-2(4) not 2 ?

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Thread starter 1 month ago
#1
(-2)^2 = 4 ?

So why is this undefined ?

Thanks !
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1 month ago
#2
I thought your question was log-2(4)? Any negative numbers with log (e.g. log-1), the answer is always undefined.

If you are asking why (-2)^2 is undefined on your calculator, then I don't know why.
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1 month ago
#3
(Original post by seals2001)
(-2)^2 = 4 ?

So why is this undefined ?

Thanks !
is not well defined on the real numbers when so the logarithm isn't either.

You need to introduce complex numbers to deal with this properly.

For instance, we can rewrite

and it is easy to be temped to say that ... but actually, this is not the only value it takes.

Using the change of base law we have

but now we can use complex numbers here.

If is a complex number, where and is the principal argument of , then we have

which means that

thus we have

for .

So, setting gives us , but if then we have infinitely many solutions.

This is why it is undefined, as there are technically infinitely many values that represents.
Last edited by RDKGames; 1 month ago
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Thread starter 1 month ago
#4
(Original post by RDKGames)
is not well defined on the real numbers when so the logarithm isn't either.

You need to introduce complex numbers to deal with this properly.

For instance, we can rewrite

and it is easy to be temped to say that ... but actually, this is not the only value it takes.

Using the change of base law we have

but now we can use complex numbers here.

If is a complex number, where and is the principal argument of , then we have

which means that

thus we have

for .

So, setting gives us , but if then we have infinitely many solutions.

This is why it is undefined, as there are technically infinitely many values that represents.
Amazing explanation !! Thank you
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