# Exponential equations

$2^x-3^{x/2}-1=0$

We can guess or use W|A to figure that $x=2$. But can we crack this problem analytically?

This problem arose out of Karim Mohamed's solution to Big Problem, small solution.

Note by John Muradeli
5 years, 5 months ago

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Do you mean finding a general solution in terms of functions?(logarithms for example) or just approximating and finding number of solutions????

- 5 years, 4 months ago

Well, general solution and a method of solving... Is there one? Like, you know, $ax^2+bx+c=0 \Rightarrow x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, and so on. Does this form of equations have a general solution? If not, that's ok. Just a solution to $2^x-3^{x/2}-1=0$ would be fine too.

Thanks

- 5 years, 4 months ago

A non-rigorous proof would be to note how

$2^x-3^{x/2}$

grows larger and larger as $x$ increases. A quick table would quickly confirm. If we let $y=2^x-3^{x/2}$, we have:

$(2, 1)$,

$(4, 5)$,

$(6, 37)$,

$(8, 175)$,

and so on and so forth.

- 5 years, 5 months ago

Not seeking for a proof here, but an analytical solution. What if we had instead

$2.71828^x-6.67384^{x/2}-360=0$?

Try making tables now.

- 5 years, 5 months ago

Can this even be done? Are equations in form $a^x+b^{x/2}+c=0$ solvable? Seems tempting to complete the square, until we realize it doesn't work like that on exponentials.

Oh, maths.

Thanks, Finn.

- 5 years, 5 months ago