A Nice Exponential Equation | Math Olympiads

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x^{1/ln(x)}=3^{1/ln(3)}
#ChallengingMathProblems #ExponentialEquations
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[seanfraser]

The equation is true for all x>0 except x=1

[JefiKnight]

Negative numbers also work if you dip your for into the imaginary world. It's weird. Inputs are real, outputs are real, but the solution is imaginary sort of.

[yttyw]

x= any positive real number (except 1) cuz of change of base formula

[popitripodi]

x^1/lnx=e^lnx•1/lnx=e.Very interesting ❤

[maxvangulik]

Can we confirm the limit as x->1 algebraically?
L = limx->1 [x^1/ln(x)]
ln(L) = limx->1 [ln(x)/ln(x)]

Because 1^♾ is indeterminate, we can use L’Hopital’s rule.

=H limx->1 [(1/x)/(1/x)] = 1
L = e
✅

[edgardojaviercanu]

It is a constant function! The answer is the domain of the function.

[sngmn]

1/ln(x) = log_x(e)
x^log_x(e) = e
e = 3^(1/ln(3)) True
For all "x" in the domain it would be true.

[americoaraya]

x^(1/log(x)) = y, where y=10 is true for all x>0 except x=1.

[sabasmoreno]

X^(1/lnx)=e. para todo x diferente de 1

[kianmath]

The sol: x is true for (0, 1) U (1, infinity)

[mathswan]

x can be any positive real numbers except 1

[angelishify]

Po definiciji: x=e^ln(x), x^(1/ln(x)), =e^[ln(x)/ln(x)]
Eksponentna funkcija je zvezna, zato velja:
Po L'Hospitalu dobimo ko gre x->1: lim[ln(x)/ln(x)] = lim [(1/x)/(1/x)]=lim[1/1]=1
Enačba torej velja tudi za x=1.

[goldfing]

To instantly find x = 3 will give the Fields Medal, won't it?

[goldfing]

x^(1/ln(x)) = 3^(1/ln(3))
Take the ln(( on both sides:
1/ln(x) * ln(x) = 1/ln(3) * ln(3)
1 = 1
for all x > 0 and x <> 1
(since ln(x) may not be zero because it is the nominator of a fraction).

[marceliusmartirosianas]

X^1/Lnx = 3^1/Lnx][ = Lnx=1=Lnx ] tada : X= 3 AcademiC Marcelius stanford university General Doctor expert