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Exponential equations

Here's something nontraditional:

\(2^x-3^{x/2}-1=0\)

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
2 years, 9 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???? Hasan Kassim · 2 years, 9 months ago

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@Hasan Kassim 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 John Muradeli · 2 years, 9 months ago

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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. Finn Hulse · 2 years, 9 months ago

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@Finn Hulse 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. John Muradeli · 2 years, 9 months ago

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@Finn Hulse 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. John Muradeli · 2 years, 9 months ago

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