Solving A Homemade Exponential Equation

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PLAYLISTS 🎵 :

[briancatanzaro]

I am happy with keeping it "real".

[vladimirkaplun]

I do not see what you see. if c<1 y^y=c^c may have 2 solutions. Hence you had to notice that as long as c>1 there is only one

[seanfraser]

Raise both sides to the 5th power:

x^(15x^15) = 2^(5*32) = 2^(5*2^5)

Using exponent rules, we can rewrite this equation like this:

(x^15)^(x^15) = (2^5)^(2^5)

The function f(u) = u^u is one-to-one for u > 1, so the only solution to u^u = 32^32 is the obvious one, u=32. Thus

x^15 = 2^5.

Taking the 15th root of both sides gives us x = 2^(1/3).

[yoav]

Nice! Time is runing and time is money😂💯

[vighnesh]

x^x = y^y => x = y is one of the solutions. There can be other solutions as well. For example. x = 1/2 and y = 1/4 are also possible solutions where x != y. How to find other solutions?

[walterwen]

Solving A Homemade Exponential Equation: x^(3x¹⁵) = 2³²; x =?
[x^(3x¹⁵)]⁵ = (2³²)⁵ = (2^2⁵)⁵, x^(15x¹⁵) = 2^[5(2⁵)]
2^[5(2⁵)] = 2^[15(2¹⁵⸍³)/3] = (2¹⸍³)^[15(2¹⸍³)¹⁵] = (³√2)^[15(³√2)¹⁵]
x^(15x¹⁵) = (³√2)^[15(³√2)¹⁵]; x = ³√2
Answer check:
x = ³√2: x^(3x¹⁵) = (³√2)^[3(³√2)¹⁵] = [(³√2)³]^[(³√2)³]⁵ = 2^2⁵ = 2³²; Confirmed
Final answer:
x = ³√2

[SidneiMV]

x³^x¹⁵ = 2^2⁵

x³ = 2 => *x = ∛2*

but if you don't believe it..

x³^x¹⁵ = 2^2⁵
x¹⁵lnx³ = 2⁵ln2
x¹⁵lnx¹⁵ = 2⁵ln2⁵
W(x¹⁵lnx¹⁵) = W(2⁵ln2⁵)
lnx¹⁵ = ln2⁵
x¹⁵ = 2⁵ => x = ∛2