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# 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.

2 years, 2 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???? · 2 years, 2 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 · 2 years, 2 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. · 2 years, 2 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. · 2 years, 2 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. · 2 years, 2 months ago