Why does e^ln3 = 3 ??

I get that they're reverse operations but am still really confused as to why the answer is 3.

Is there a method so I could see how exactly they cancel to get 3.


Posted from TSR Mobile

I get that they're reverse operations but am still really confused as to why the answer is 3.

Is there a method so I could see how exactly they cancel to get 3.


Posted from TSR Mobile

I get that they're reverse operations but am still really confused as to why the answer is 3.

Is there a method so I could see how exactly they cancel to get 3.


Posted from TSR Mobile

It's basically the same reason as to why $\sqrt{a^2} = a$, or $\sin(sin^{-1}(x)) = x$.

If you said $f(x) = e^x$, then we have $f^{-1}(x)=\ln x$.

Then we know that $f(f^{-1}(x)) = x$ (by definition of them being inverses to each other), and so $f(f^{-1}(x)) = f(\ln x) = e^{\ln x} = x$.

Well first of all this question wouldn't make sense without knowing the definition of the function f(x)=ln(x) just like 1+1 doesn't make sense without knowing what + means. Therefore we need to define ln(x). This is defined as the inverse of g(x)=e^x. So by definition f(x) satisfies g(f(x))=x, i.e. e^(f(x))=x, which is what the question wants since f(x)=ln(x) (then we just put x=3).

So this question follows from the definition of f(x).

In fact, all of maths follows from the definitions, just like you only need the definitions of geometry and the obvious axioms (which mostly can be taken for granted without stating, because they're obvious). But the fun thing about maths is that there are also deductions; this isn't available in a subject such as languages. In languages, a beginning student must study very simple exercises that bear no resemblance to the actual subject of languages, with its beauty, depth and elegance. He must study trivialities such as: convert "the dog barked" into latin, and similar rubbish. Whereas on the other hand, a mathematics student, even a beginner, can be given exercises early on that are as close in originality and spirit as the greatest theorems of his subject. Such are for example many olympiad problems in geometry, many of which I am sure even the ancient greek geometers would have been proud of.

This is why mathematics is such a beautiful subject. Indeed, the whole subject stems from deductions from the axioms/definitions, just like a tree stems from the seed and its roots. Edit: This is also why it has the greatest proportion of cranks (physics is second), because they think they are doing original work by discovering deductions, but more often than not their "deductions" are simply restatements of a fundamental principle or well-known theorem (while the rest usually fall into the category of "olympiad problem").

ln3 basically says what number do we raise e by to get 3
So e^ln3 is raising e to that power - to the power that we raise e by to get 3. So we get 3

Actually a good response rather than the smug stupid answers other people give.

from a teaching point of view I agree.
from a mathematicians point of view this is an incorrect response.

There is nothing mathematically incorrect about what I said. It may not be as technical as a 'mathmetician' would like but what I said can help people to understand it.

Isn't that exactly what I said?

No you said it was an incorrect response from a mathmetician a point of view. It's not incorrect from any point of view - it can't be because it is not incorrect

It's basically the same reason as to why $\sqrt{a^2} = a$

I am not going to engage into a debate or an argument with you ...
I sincerely wish you all the best

This is incorrect.

$\sqrt{a^2} = |a|$

I am sure brittanna knows that, but I am sure he did that for the sake of simplicity, given the OP did not understand why e^ln3 = 3

I am not going to engage into a debate or an argument with you ...
I sincerely wish you all the best

No you said it was an incorrect response from a mathmetician a point of view. It's not incorrect from any point of view - it can't be because it is not incorrect