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Help Please (1 Viewer)

vds700

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How would you solve |2x|=2^x and what are the answers.
you can break it down into cases (by observing the graphs of y = |2x| and y = 2x)
for x > 0:
2x = 2x
x = 2 (by inspection)

for x < 0
2x = -2x...not quite sure how to solve this one
 

lolokay

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x = 1 or 2 for the positive case

not sure if the negative case is possible to solve, apart from doing approximations
 

Aerath

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Approximations should be fairly simple, since there is only a positive gradient. Answer is about -0.38.
 
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Manboobs

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ahh, well to get that third solution is pretty sucky. Who gave you that question?

Also i really don't think you should approximate answers for the hsc. The only way i can see answering that is to use the lambert W function. Your answers are very close.

Lambert W function - Wikipedia, the free encyclopedia

2^x = 2x assuming x<0
1 = 2x/(2^x)
1 = 2x*e^(-x*ln2)
(-ln2)/2 = (-ln2*x)*e^(-x*ln2) this is in lambert form
-ln2*x = W(-ln2/2) W is the lambert W function


x = [-W(-ln2/2)]/ln2 this will give you your result

it might be worth noting that this uses the 3rd solution to x^2 = 2^x. Solutions 2, 4, -0.7666646959....

where x ln|2| = ln |-0.7666646959|

x = ln(0.7666646959)/ln2 = -0.383332348

there's probably a reason why but this isn't the place. I suspect there's a really easy solution using complex numbers. Sorry for babbling.
 

Aerath

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Yeah, definitely not in the 2U syllabus. =P
 

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