How to solve the exponential equation x^ln(x)=2 by using logarithm

[@moeberry8226]

No solution for question 2. Your function is actually just a straight line y=e with a little domain restriction so the line y =2 is always below it and most importantly parallel to it and therefore never intersect. You have two holes one at (0,e) and the other at (1,e). And obviously the domain is (0,1) u(1, infinity)

[@Luke-rb8dh]

Proving that this is a straight line, you can do x^(1/lnx)=x^(lne/lnx) = x^(logxe) =e

[@twelfthdoc]

For the given equation, we can take the log of both sides use logarithm rules to bring the 1/lnx power down to the front. Unfortunately the product of lnx and 1/lnx causes cancellation down to just 1 and we are left with the equation 1 = ln2 which is a contradiction. Therefore, the equation x^(1/lnx) = 2 has no real solutions (despite how many times an engineer will tell you e and 2 are close enough).

[@johnchessant3012]

x^(1/ln(x)) asks what to the power of ln(x) equals x. But we know that e to the power of ln(x) equals x, so x^(1/ln(x)) = e, for all x.

[@speaker_of_the_house66]

For all x > 0, x =/= 1. You still have to make sure the function is defined at that point

[@ronycb7168]

Can you make a tutorial on how to graph tricky fuctions like those.. Thanks!

[@اميرمحمد-ث4ك5خ]

in last step you can bring the roots up and delete the e with the ln to get 2 in root and -1over 2 in root

[@darkahmedp]

This is a fantastic channel iam from egypt and have 12 and half years old and do calculus with this channel, thank you very much,with arabic شكرا جزيلا.

[@dihey4689]

X to the 1/ln X is e as x approaches infinity and we know by the fundamental theorem of engineering that e = 2 so X is infinity

[@leonardobarrera2816]

Thanks, actually, I need that for my homework You are amazing!!!!

[@gregorkowalski1032]

Love it.. But best of all is him playing in perfect way with red and black pens... Absolutely fantastic 😍😍😀

[@odysseus9672]

Even better: prove that if we define the binary operator ~ by x~y = x^ln(y), then it is commutative (i.e. x~y = y~x). Likewise, prove that x~(y~z) = (x~y)~z (i.e. it's associative) and x~(y*z) = (x~z)*(x~z) (i.e. it's distributive across multiplication). From this, show that there is an infinite series of commutative and associative binary operators and that each one distributes across the binary operator that comes before it in the sequence.

[@ultrio325]

Assuming x is in the domain of ln, lnx/lnx=ln2=>1=ln2=>False. Thus, our assumption is false, and x is not in the domain of ln. Therefore ln(x), 1/ln(x) and x^(1/ln(x)) are all undefined, meaning the given cannot be true, i.e. the equation has no solutions.

[@Spookyy6]

Bonus question: let X = e^a Substitute into the equation: (e^a)^(1/ln(e^a)) = 2 (e^a)^(1/a) = 2 By rules of exponents, that is equal to: e^(a*1/a) = 2 e^1 = 2 or e = 2 Since e is not equal to 2, the equation has no solutions.

[@chipan9191]

For x^(1/ln(x)), this can be reduced by putting it in the form (e^ln(x))^(1/ln(x)) which is e^(ln(x)/ln(x)) which reduces to e¹ which is e. Only caveat is that x>0 & x≠1, but the only real solutions at all points if this expression is e. That means it cannot equal 2.

[@andresromero488]

I saw some answers saying that x^(1/ln(x)) = 2 has no answer. And thats true, but almost all of them are incomplete. They only consider the real numbers, but, what if there was a complex answer? In complex numbers it is not always true that ln(a^b) = b*ln(a). For example, e^(3*pi*i) = e^(pi*i) = -1. And if ln(a^b) = b*ln(a), that implies that 3*pi*i = pi*i and obviously that's not true. And most of the answers I saw use this property. One solution that I think is more complete is this: For complex numbers, by definition, a^b = e^(b*ln(a)), where ln(a) means the "principal value of the logarithm" (Ii is similar to the case of square roots of positive numbers, where the "principal value" is the positive one). So for all numbers for which the function f(x) = x^(1/ln(x)) is define (all complex numbers except 0 and 1), the value of it is: x^(1/ln(x)) = e^(1/(ln(x)) * ln(x)) = e^1 = e Now we can afirm that the function f(x) is constant and its value is e. Therefore, there is no x such f(x) = 2. So there is no solution.

[@greenforest9432]

Power of ln(X) on both sides. We have then : X = 2**ln(X). Then the ln gives the expression : ln(x) = ln(x)ln(2). X =/= 1 because we cannot divide by 0. So we can simplify by ln(x) in both sides : 1 = ln(2) which is preposterus. Then the equation has no solution.

[@a_man80]

I have an equation, can you solve this: x•(x^x)=x+(x^x)

[@fedebic5443]

That last x^(1/lnx) is just a fancy way of writing e, still very cool video as always!

[@dhwaneelkapadia3265]

from the second question, how would we solve (lnx)^x = n where n is any number

[@orlandot6]

How do you know that the graph is gonna look like that?

[@Kanal263]

No solution for question 2, because ln on both Sides is Equal to 1 = ln(2) e = 2 and thats Wrong.

[@guitarttimman]

e^sqr(ln2) = x

[@felina3581]

How did you graph that in your mind

[@max-hi1bk]

I dont understand the first step, how can lnx to the power of lnx be lnx times lnx? Like 4^4 isnt 4x4

[@teamzumali2914]

I used lambert function to solve the first one, does it work?

[@temujnbn9456]

I come up with this equation: logx(5x)=2x/5 Can you solve for me

[@SuperYoonHo]

Thanks MR BPrP SLOW

[@darkahmedp]

Phi,it has no solve, because we get in for other sides and with some maths we will get in(2)=1,that's mean 2=e.

[@ruilongsheng2845]

lnx=√ln2, x=exp(ln2)

[@OBGynKenobi]

Why is lnX*lnX= (lnX)^2, but sinX*sinX = (sin^2)X ??

[@ezecattalin8]

e^(±√(ln(2))*i)

[@colinjava8447]

Got it right, I'll try the harder one.

[@spektator5418]

haha nice ending question I tried taking random samples before trying anything and the answer was always e then I tried graphing the function and its litterally |y|=e there is no solution!

[@SuperYoonHo]

last question 1=not ln2

[@hac1_ahmet]

ca you creat a dısord server

[@timtsai6860]

右邊那隻是茶里的玩偶嗎？

[@adelitogd]

YOO I ACTUALLY GOT THIS CORRECT LETS FRICKING GOOOO

[@sohampinemath1086]

Fun one

[@aug3842]

x^(1/lnx) = 2 ln(x^(1/lnx)) = ln2 1/lnx * lnx = ln2 1 = ln2 however 1 ≠ ln2 => no solutions

[@mrlitty8330]

Wait what? For the question at the end of the video, I got 1=2 because I tried taking the ln of both sides but the left side canceled out to 1.

[@Ni999]

1=ln2 means that it's a trick question, there's no solution. Remember - x = e^lnx so x^(1/lnx) = (e^lnx)^(1/lnx) = e for x>0 and x≠1 so e ≠ 2.

[@p12psicop]

x^(1/lnx) = e

[@leeshaocheng239]

I think the answer might be complex solution

[@muhammedfuadpt5137]

There will not be a solution for the last question

[@finmat95]

wow...

[@fatihsrk]

W(2) :-)))

[@NurHadi-qf9kl]

Ln x=1 ; x=e

[@social6332]

very funny question.

[@velk_wanggoudan]

ez one babe

[@muhammedfuadpt5137]

First

[@leonardobarrera2816]

Oh, lots of luck sir!!!!

[@godturtle6274]

the only solution would be e so 2 doesn't worl

[@starpawsy]

If I cant solve it in two minutes, I move on to another video. Bye.

[@douglasdeoliveiracardoso9345]

I didn't understood how e^ln=1

[@andresromero488]

By definition of the natural logarithm. The logarithm of a number x to the base b is the exponent to which b must be raised, to produce x. That is b^(log_b(x) = x). The natural logarithm is the logarithm in base e. So ln(x) actualy means log_e(x). That means that is the exponent to which e must be raised to produce x. In other words, e^(ln(x)) = x

[@douglasdeoliveiracardoso9345]

@@andresromero488 didn't get yet, I already knew about that definition. what I want to know is the proof that it is right, the derivation

[@guitarttimman]

Notice that I didn't need to watch the video.

[@sourovdas7883]

heha

[@Claytonbeastboy]

No solution. When you cancel everything put you get 1 = ln(2)

[@Claytonbeastboy]

Oh holy crap I guess that's just e huh

[@at7388]

Why are you boring us with elementary school assignments?

[@СВЭП-и4ф]

I solved by substituting x = 2^t, which leads to 2^(t^2) = e, => t = (log2(e))^(1/2), => x = 2^(log2(e))^(1/2), interesting that plugged into calculator both my answer and answer from video lead to 2.29918476743