I can visualize the exponential function with the graph $y = e^x$, but I can do that for almost any function.
In addition to its graph, the function $f(x) = x^n$ can be visualized as the volume of a box with sides of length $x$ in n-dimensional space, and the trigonometric functions can be interpreted as side lengths of certain right triangles.
Is there a similar geometric interpretation of the exponential function?
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This answer is not about $e^x$, but about the closely connected hyperbolic functions $$ \cosh x=\frac{e^x+e^{-x}}{2},\quad\sinh x=\frac{e^x-e^{-x}}{2}. $$ A little algebra shows that $\cosh^2x-\sinh^2x=1$. Thus, $(\cosh x,\sinh x)$ are points in the hyperbola $x^2-y^2=1$; then $\cosh x$ and $\sinh x$ are the legs of a right triangle whose hypotenuse is the segment joining the origin and the point with coordinates $(\cosh x,\sinh x)$.
A different interpretation is the following: Imagine that you are running away fron some fixed point $O$. If your speed at each moment is equal to the distance to the point $O$, then your speed will be $C\,e^x$ for some constant $C>0$.
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If what you're looking for is a visual way to understand it, while a geometric interpretation might be useful, the most intuitive way to understand $e^x$ for me was stuff like population dynamics.
If you've ever heard the phrase "exponential growth" in relation to a population of something (bacteria, fish, wildlife, etc), it refers to the idea that each change in the population is linearly proportional to the population at the last time step -- assuming no outside influences in any way hinder or alter the population dynamics.
It stems from what I think is probably the simplest non-trivial ordinary differential equation there is:
$y'(t) = y(t)$
That is, there exists some function $y(t)$ such that $y(t)$ is the same function as its first derivative (up to a constant). The only function satisfying this criteria is $e^x$.
The implication may not be immediately obvious, so I hope you'll forgive the example.
If a population of fish is accurately described by $y$, then the larger my fish population is the faster the population will grow -- exponentially so. At each iteration, assuming a fixed average spawn-rate-per-fish, then the new fish added to the population will be a percentage of the size of the current population, and since the population grows every round then the increase and the population both grow at about the same pace.
This is true of financial investments as well that are strictly based on a fixed interest rate.
Of course, these are simplistic models. Fish populations don't really grow exponentially and financial investments don't really produce exponential growth.
-Brian
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Here is a simple geometric construction which allows you to visualize $f(x)=e^x$.
Take two objects on a flat level surface, one at the point $(0,0)$ and the other at $(0,1)$, that are connected by a piece of string. Slide the top object along the horizontal line $y=1$. The horizontal displacement of this object is given by $x$, and the trajectory of the bottom object (the path of least resistance) is called a tractix.
After sliding $x$ units horizontally, draw the line segment of length $1$ which is parallel to the string and which contains $(0,0)$ as one of its endpoints. The second point of this line segment lies on the unit circle. Draw a line through the circle endpoint and the point $(1,0)$. The $y$-intercept $b$ of this line is equal to $e^x$, and the point $(x,b)$ belongs to the graph of the exponential function $f$.
Here is an animation I made using GeoGebra: https://youtu.be/zzvwGl9WpX8
There's a geometric interpretation of the natural log. From the definition
$$ \log x = \int_1^x {1 \over t} \: dt $$
we see that the area between the "standard" hyperbola $xy = 1$ and the horizontal axis between $1$ and $x$ is $\log x$.
So, turning this around, the line $x = e^t$ is the vertical line such that the area between $x = 1$ and $x = e^t$, between this hyperbola and the $x$-axis, is $t$.