Find the image of the line $y=mx$ under the exponential application.

Question

Find the image of the line $y=mx$ under the exponential application.

I'm not sure if this question will make sense since I've translated it from french to English, but I'm not sure how to start this...

Any help or guidance would be much appreciated.

Thank you in advance!

Answer 1

I suppose that $m\in\mathbb{R}$ is a constant. The image-set is the set $E=\{e^{x+imx}: x\in\mathbb{R}\}=\{e^x\cdot e^{imx}:x\in\mathbb{R}\}=\{e^x\cdot(\cos(mx)+i\sin(mx)):x\in\mathbb{R}\}=\{\big{(}e^x\cos(mx),e^x\sin(mx)\big{)}: x\in\mathbb{R}\}$

Therefore the image set is a logarithmic spiral. Pick any $m$ you like and graph this set on any parametric graph calculator, you will be convinced!