solving an exponential equation

[chrislav]

solve the following equation:
$$x^{x^4}=64$$ in real Nos

Put $$x^4=y$$ ,then $$y^y= (x^4)^{x^4}=(x^{x^4})^4=64^4=2^{24}$$
And taking logs we have :$$ylog_2{y}=24$$ which implies that y=8

And hence $$x=2^\frac{3}{4}$$ or $$x=-2^\frac{3}{4}$$
Is that correct?

 

[Steven G]

Why not check it for self? That is what I did. Do both values for x work?

 

[chrislav]

Why not check it for self? That is what I did. Do both values for x work?

The point is if the whole proof leading to a solution is correct
yes you right
it seems that the positive value works

 

[blamocur]

The point is if the whole proof leading to a solution is correct
yes you right

I haven't noticed any proofs there, but I don't know if your assignment requires them. Myself, I don't know how to get from $y\log_2 y=24$ to $y=8$ except by a lucky guess. But you could improve your solution by proving that there are no other solutions to this equation.

it seems that the positive value works

What about the negative value?