Graph of a logarithmic function

Question

When deriving the graph of a logarithmic function, why does the graph have to be reflected after interchanging the x and y axes? Is it possible for the axes of a graph to be changed? If so why?

Answer 1

Because

$$y=\log x\iff x=e^y.$$

If you are able to plot the graph of the exponential, then you can draw that of the logarithm by swapping the axis.

This works with all direct and inverse function pairs, such as $$y=x^3\iff x=\sqrt[3]y$$ and so on.

Answer 2

Attempt:

1)Consider $y=f(x)$; or $(x,f(x))$.

Example: $y=e^x$; or $(x,e^x);$

2) Reflection about the line $y=x$:

Point $(a,b) \rightarrow (b,a)$.

Example: $(x,e^x) \rightarrow (e^x,x)$.

3) We are looking for something like $(x,F(x))$.

4) How to get from $(e^x,x)$ to $(x,F(x))$?

5) Set $X:=e^x$ ($>0$).

The inverse function:

$Y=\log X =x$; or instead of

$(e^x,x)$ we can write

$(X,Y) = (X,\log X)$ where $X >0$.