help with function composition

[Henry needs help]

at my school I was tasked with explaining something, but I couldn't understand

Function Composition: "log" and "e^x" (how will be inside the other)



_{sorry for any english mistake
I for my desperation came looking for forums from other countries}

 

[Steven G]

I have two questions. What does 'how will be inside the other' mean?
Log really has no meaning. You can compute log(2), log7, log(2/3) and even log(e) but log has not real meaning so I am a bit confused.

Possible you just need to compute log(e^x) ???

If this is correct then you should use the fact that \(\displaystyle \log_a{b^c}= c\log_a{b}\)

You can even go further if by log you mean log_e or ln

 

[Dr.Peterson]

at my school I was tasked with explaining something, but I couldn't understand

Function Composition: "log" and "e^x" (how will be inside the other)

_{sorry for any english mistake
I for my desperation came looking for forums from other countries}

One way to deal with language difficulties is to show us the exact, entire problem in the original language, and also attempt to translate it.

In addition, if you tell us as much as you can about what you are asking, using more words than necessary, then the redundancy may help us understand.

Also, define your terms. When you use "log" by itself, does that mean the natural log (base e), or something else such as base 10? Anything that may be different in your culture needs to be stated.

But if you mean [MATH]\log_e(e^x)[/MATH], then this is the composition of inverse functions, and the result is just [MATH]x[/MATH]. The composition in the other direction is [MATH]e^{\log_e(x)} = x[/MATH].

 

[Otis]

Hello Henry. The base-e logarithm and exponential functions are inverses of each other. That means each function undoes what the other does.

Here is a simpler example of inverse functions.

f(x) = x² (the squaring function)

g(x) = √x (the square root function)

Let's start with the squaring function f. If we put 4 in, then 16 comes out.

f(4) = 4² = 16

If we put that 16 into the square root function, then 4 comes out. The square root function undoes what the squaring function does.

g(16) = √16 = 4

Let's try starting with function g. If we put 9 into the square root function, then 3 comes out.

g(9) = 3

If we put 3 into function f, then we get 9 back.

f(3) = 9

The functions x² and √x are inverse functions. Each undoes the other.

If we compose f and g, the output will always be the input.

f(g(4)) = 4

g(f(9)) = 9

So we write f(g(x)) = x = g(f(x))

If any number x goes into either composition, then the same number x comes out.

e^x and log_e(x) are also inverse functions. Each undoes the other.

If you compose them, the input and output will be the same number.

If you haven't yet learned about function compositions (or the definition of a logarithm), then you ought to study those topics first, to see more examples and explanations. Cheers

?

 

[pka]

at my school I was tasked with explaining something, but I couldn't understand
Function Composition: "log" and "e^x" (how will be inside the other)
_{sorry for any english mistake
I for my desperation came looking for forums from other countries}

Because of your statement about English, I have no way of judging what grade level you are on..
Be that as it may, I understand that few helpers on this site will agree with me on this.
I think tat within ten years all calculators & computers will use \(\displaystyle \log\) to mean base \(\displaystyle e\).
So \(\displaystyle \log(e^x)=x\) & \(\displaystyle e^{\log(x)}=x\)

If you have \(\displaystyle 6^x=18\) then \(\displaystyle x=\dfrac{\log(18)}{\log(6)}\)

If \(\displaystyle \log(x+1)=65\) then \(\displaystyle e^{\log(x+1)}=e^{65}\) or \(\displaystyle (x+1)=e^{65}\)