Solving An Exponential System

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[MichaelRothwell]

Nice! This is basically the parametric solution of xʸ=yˣ for x, y>0 and x≠y that you get by using the substitution y=tx in the particular case where t=5. A clever way to make it fit in a short!

In your case you should really specify that x, y>0. (x≠y is then automatic given that y=5x).

First, because you need to exclude the obvious solution x=y=0.

Second, because negative solutions can also arise for this type of equation.

For example, x²=2ˣ has two obvious positive solutions x=2, x=4, but also has a negative solution, as you can see by graphing the functions.

The negative solution is
x=-W₀(ln √2)/ln √2,

[justtheletterV]

X can be (5)^(1/4), -(5)^(1/4), (5)^(1/4)i, or -(5)^(1/4)i

[SkeletonKingYoutube]

You can divide by x since x=0 is not a solution

[salmankhan-bwdo]

Avobe equation only true when x=y
Or you have to specify that x≠y and x, y=0 then you can solove these equation....

[armacham]

skipping some important steps to keep it short. like showing that x=0 is not a solution and showing why x=-quadroot5 is not a solution. I mean if someone is watching these math shorts, they probably already know that.