Find easy root of two.
The equation below has three roots. Find the one that can be given without using digits to the right of the decimal point. As for many problems here on brilliant, it is quite easy if you attack it the right way.
2 0 1 4 x = x 2 0 1 4 .
The answer is 2014.
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5 solutions
What would be the other root? And also, wouldnt it be easier to use the logarithms in base x?
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I hadn't tried using base x log; it simply occurred to me not to do the usual which is to take natural logs. Having done that, I was charmed by the symmetry of that equation shown in the line beginning with the word 'rearrange'.
Must get back to you, perhaps this afternoon, about the other part of your question.
Actually there are two according to Wolfram-Alpha but I can only do the algebra to get one of them. Let me proceed slightly more generally, to solve a x = x a . The solutions are in terms of Lambert's W, therefore, let me take natural logarithms of both sides of this equation and manipulate it much as I did above, to obtain:
a ln a = x ln x
The usual way with applying Lambert's W, I understand, is to attempt to rearrange an equation to a form like c = f ( t ) e f ( t ) . The rhs of the equation above can be written ln x e − ln x . Multiplying both sides of the equation by -1 it's now in the desired form and the Lambert W can be applied as an operator to obtain solutions.
Except that I can't see how to get the second real solution. My lousy algebra perhaps!
To make the two sides equal, we need to make the two expressions equal. 2014 does a pretty good job of that.
The other two solutions are near 1 . 0 0 3 8 and − 0 . 9 9 6 2 4 4 by the way, if anyone cares
Pretty straightforward that x = 2 0 1 4 since 2 0 1 4 x grows faster than x 2 0 1 4
What? How can it grow faster when they are the same number?
ln(2014^x)=ln(x^2014) x.ln2014=2014.lnx x/lnx=2014/ln2014 So the solution of this problem is x=2014.
take y^x=x^y. xIny=yInx. rearranged, Iny/y=Inx/x. deducing, y=x.
from here, is it (2014)^2=2^(2014) and so on?
Note that ln y / y = ln x / x does not imply that y = x .
For example, ln 4 / 4 = 2 ln 2 / 4 = ln 2 / 2 .
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can you give the two other roots using pre-college algebra??
2 0 1 4 x = x 2 0 1 4
Take logarithms of both sides (what else!): x lo g 2 0 1 4 = 2 0 1 4 lo g x
Rearrange: x lo g x = 2 0 1 4 lo g 2 0 1 4
Conclude: x = 2 0 1 4 .
Adapted from a question on quora.com.