Understanding Exponential and Natural Log Rules: Is this Simplification Correct?

[thomas49th]

Is it true that

e^{2(2x+ln2)} = e^{4x}e^{2ln2}

I can't see how that is true? According to a paper mark scheme it is.

Can someone clarify
Thanks :)

 

[CompuChip]

First open up the brackets:
e^{2(2x+\ln 2)} = e^{4x + 2 \ln 2}
Then use a rule for e^{a + b} which you (should) know.

Actually, you can even simplify it further to 4 e^{4x}.

 

[thomas49th]

is this the same with all powers or just the exponetial function... hang on, is it a function?

 

[CompuChip]

Well, the first simplification (the one you asked about) works in general. If x is any number and a and b are two expressions, then
x^{a + b} = x^a x^b
so in particular it works for x = e, a = 4x and b = 2 ln(2).

The further simplification I spoke about just works because ln[..] is the inverse of exp[..] = e^[...]

 

[thomas49th]

yeah i can see the second one.
The first one makes sense now. Yes. I just didn't see it with the 'e'

cheerz :)

 

[CompuChip]

So the lesson to be learnt, perhaps, is that

"e^x"​

can be viewed both as the function "exp" evaluated in (some number) x, or as the number e \approx 2,7\cdots raised to the power (some number) x and that how you view it depends on the context (e.g. when differentiating it, one should view it as a function; when using simplification rules like here it's easier to just view it as an exponentiation).