[–] 7 points8 points

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ex and ln(x) are inverses of one another. So you could rewrite (ex , x) as (x, ln(x)).

 e^y = x
ln(e^y) = ln(x)
y = ln(x)


Another way to look at it is to plot ex and ln(x) on the same graph. Notice they reflect over the line y=x. This isn't a formal way to look at inverses, but it serves for this purpose.

[–] 0 points1 point

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A much simpler example would be y=x+1 and y=x-1.
They are inverse functions and reflect about y=x so, if you were to 'flip the axes' so to speak of either function then you would be left with the other.

Also, as actingSmart mentioned, the curves intercept on the line y=x in your example.
It should be obvious why.

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It might help to imagine it as a parametric curve. Let f(t) = (a,b), where a(t) = t, b(t) = et.

Now, when you graph f(t) in the xy-plane, notice that for each increase of t by 1, x increases by 1, and y increases by e. What we essentially get is that the graph of f(t) is analogous to the graph of y=ex. In general, a parametric curve defined as (t, [some function of t]) looks identical to the plot of y=[some function of x].

Now, flip them! Using the idea that (t, [some function of t]) looks like the plot of y=[some function of x], what might we be able to induce about ([some function of t], t)? If we plot g(t) = (b, a), every increase of t by 1 increases x by e and increases y by 1. This resembles the inverse of the exponential function, which is the natural logarithm function.

So, what's happening when you switch (x, ex) to (ex,x), you're taking the inverse of the function ex, which is the function ln(x).

[–] 0 points1 point

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We have points (x, y)

So for (x, ex ) what function is that? What function takes in x gives you ex ?

Well, it would be y=ex , you plug in x and get out ex

ln(x) is the inverse of ex , it undoes what ex does: ln(ex ) = x

Now lets look at the points (ex , x)

What function will take in ex and spit out x?

We said earlier that ln(ex ) = x , so the points (ex , x) are just the function ln(x)!

[–][S] 0 points1 point

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Hi, Thank you all for the responses. It will take me time to absorb the material. Slow learner here.
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