Inverse Functions: Logarithms & Qualitative Conclusions

[HallsofIvy]

One more time and then I'm going to drop this: there is no unique way to divide a formula into two functions. There might be a "simplest" way that seems obvious but that is not a mathematical property.

 

[Byrgg]

One more time and then I'm going to drop this: there is no unique way to divide a formula into two functions. There might be a "simplest" way that seems obvious but that is not a mathematical property.

So then how were you able to say that b^{log_b(x)}= x is saying that b^x and log_b(x) are inverse functions? Later in the thread in which you said this, it seemed this was taken from the definition of the logarithmic function, however, this is what you said early on in that thread:

What you are asked to prove is that b^{log_b(x)}= x , which is precisely saying that bx and logb(x) are inverse functions. HOW you would prove that depends on what definitions of logb(x) and bx you are using.

I am confused by this, since you yourself have said that there is no unique way to divide an equation into two separate functions. However, here it seems as though you have done this. Would you, or someone else be able to clear this up for me?

 

[Office_Shredder]

byrgg, it's like this. Let's say you want to show that 12 is even. The simplest way is to say:

12=6*2, which is even by definition. Or you could do this:

12=4*3
4=2*2 so 12=3*2*2 which is even by definition

While this is a trivialized case, you can see a number of important points here:

1.) There were multiple ways of breaking up the number 12.

2.) One way was three times as fast as the other way in terms of solving the problem

It's the same thing with b^log(x)


For the second part:
Again, the problem makes it clear that you are looking at two functions, b^x and log_bx While you're able to break it up in different ways than that, it's the easiest and simplest way to do it.

Now, the reason why those two functions are best, is that b^x and log_bx are well known inverses, and you're attempting to prove that your function b^{log_b(x)}= x
is composed of a function and its inverse (because it equals x). It's composed of an infinite set of functions and their inverses, but b^x and log_bx are the easiest, simplest, and most well known inverses.

And the point of the statement was to ask how you have defined b^x and log_bx, because the definition can vary widely (log can be defined as the inverse of an exponent, for example. This would make your life very simple)

 

[Byrgg]

For the second part:
Again, the problem makes it clear that you are looking at two functions, b^x and log_b(x) While you're able to break it up in different ways than that, it's the easiest and simplest way to do it.

You said that the problem is showing you two functions, whereas HallsofIvy said that the problem consists of only one function. Could you clarify this?

 

[Byrgg]

Someone please respond.