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.

#Algebra

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Hasan Kassim - 6 years, 9 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

John M. - 6 years, 9 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.

Finn Hulse - 6 years, 9 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.

John M. - 6 years, 9 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.

John M. - 6 years, 9 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$