Q. Logs and Exponentials (1 Viewer)
- Thread starter ezzy85
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Constip8edSkunk
Joga Bonito
let e<sup>x</sup> = u
y = 1/2 (u - 1/u)
2uy = u<sup>2</sup> - 1
u<sup>2</sup> - 2yu - 1 = 0
u = [2y +/- (4y<sup>2</sup> + 4)<sup>1/2</sup>]/2
e<sup>x</sup>=y +/- (y<sup>2</sup> + 1)<sup>1/2</sup>
x=ln[y +/- (y<sup>2</sup> + 1)<sup>1/2</sup>]
=ln[y + sqrt(y<sup>2</sup> + 1)]
y = 1/2 (u - 1/u)
2uy = u<sup>2</sup> - 1
u<sup>2</sup> - 2yu - 1 = 0
u = [2y +/- (4y<sup>2</sup> + 4)<sup>1/2</sup>]/2
e<sup>x</sup>=y +/- (y<sup>2</sup> + 1)<sup>1/2</sup>
x=ln[y +/- (y<sup>2</sup> + 1)<sup>1/2</sup>]
=ln[y + sqrt(y<sup>2</sup> + 1)]
It's also interesting to note that this function is called the hyperbolic sine (or sinh) i.e. y = sinh x (that's what the "hyp" button on your calculator is for). It turns out this function is quite useful in stuff like engineering etc. If you do maths at uni, you'll study more about functions like sinh, cosh, tanh etc