Solving a Nice Exponential Equation | x^{x+1}=81

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x^{x+1}=1
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Finding one solution was easy. Showing that this solution is THE solution is TERRIFIC! 👏🏽👏🏽👏🏽

jotamedina

0:28 Lamberts W function wont work, because the only thing we can do is to ln both sides, but when we do that we get just (x+1) ln(x) = 4 ln(3), it may be seem like x=3 but since we will go theoretically we cannot use the W function because of the rules of ln; we have the formulas for ln(xy), ln(x^y), but we dont have the formula ln(x+y); maybe we can use the approximate forms using the Taylor series but the Taylor series will also have the "x", so it also wont seems to work (And also wolframalpha's library containts the lambert W function but when we put this equation to wolfram, we cannot get a nice form with Lambert's W, we will just get a numerical reel solution)

nokta

For exponential functions like this, it’s usually helpful to find the prime factorization of the constant. We have x^(x+1) = 3^4, which immediately gives the solution x=3.

We show that this is the only solution. Let f(x) = x^(x+1), which is only continuous for x>=0. Then
f’(x) = x^(x+1)[lnx + 1 + 1/x]. You can show that this function is always strictly positive (I’ll let the comment reader figure this out). Thus f is strictly increasing and thus monotonic, so x=3 is the only solution.

seanfraser

We haven't figured if there are complex number solutions. I'll think further. Another teacher did a similar problem and had the almost same graph but included for negative x the -81/(|x|) third quadrant hyperbola g(x) ... but you sort of showed rampant x^x for 0 < x. The question remains there were or were not (a + ib)^(a + ib) = 81/(a + ib) complex x = a + ib solutions.

lawrencejelsma

When you see the dots that seem to follow a curve like that, it's showing the results of the function that have no imaginary component. If Desmos had the ability to map the complex plane onto the Z axis, it would actually form a continuous spiraling function.

Phryj

Like every other viewer, it took me two seconds to see that x = 3 is the answer, although Sybermath's more analytic approach is far better than just guessing. By the way, I truly despise the W function, and I'm happy that it was rejected early on in the video!

weylguy

Hi! I loved how you solved this equation... Please i've some equation, that is dificult to solve. So, can you give some Help please? Equation: 7^x +1=8^x or 2^x + 1=3^x.

UIS_CHANEL

Re-writing the equation as 81 = xlnx(exp(xlnx))/lnx = W0(xlnx)/lnx, where W0 is the principal branch of the Lambert W function, which has the well known properties of being real, single-valued and strictly increasing for x >= 1/e, we conclude that that xx^x = 81 has at most one solution. We note that x = 3 is an obvious solution, so it must be the only one.

johnnath

Me having problems putting 3 in equations 😂

UNIVERSALDEVIL

the answer just popped into my head without solving it. And I'm no good at math.

alyoshakaramazov

You take f(x)=(x+1)*ln|x|=4*ln3, find first and second derivatives, then you can conclude that f(x)<=0 for x<0 therefore there is no solution.after that you check for x>0 finding that f'(x)>=f'(1)=2>0 therefore f is increasing for x>0, so the only solution for the equation above is the obvious x=3

vaggelissmyrniotis

But is there a way to find the solution x=3 mathmatically? I was struggling finding the solution for x^x = 81/x.

Shadi

So...x=3 is an obvious solution, and x^(x+1) is an increasing function. End of story.

rickdesper

just by intuition, I would have said x=3, then aguing that x^(x+1) is always growing, I would have said that it is also the only solution

riccardoguareschi