Help Please (1 Viewer)

[Air Jordan]

How would you solve |2x|=2^x and what are the answers.

 

[vds700]

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 = 2^x)
for x > 0:
2^x = 2x
x = 2 (by inspection)

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

 

[lolokay]

x = 1 or 2 for the positive case

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

 

[Aerath]

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

 

[Air Jordan]

K, thanks everyone

 

[PC]

x = -0.3833, 1, 2

 

[Manboobs]

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]

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