Why Does e Raised to the Natural Log of 2 Equal 2?

[rudders93]

Homework Statement

Hi, I was wondering why: e^ln(2) = 2. I'd just like to see how to get this result instead of just having to memorize it

Thanks!

 

[LCKurtz]

Take the natural logarithm of both sides and remember that two positive numbers whose logs are equal must be equal.

 

[Tobias Funke]

It's simply a matter of definition, nothing more. You need to fully understand inverse functions and the conditions under which a function has an inverse. e^x takes any number and outputs a positive real number (I'm assuming you're restricting your attention to real numbers) and it's also one to one and onto the positive reals, so it has an inverse function that takes any positive number and returns some real number.

So you have some number x and you exponentiate to get e^x. Now, every positive number y is the result of e raised to some power (it's not a proof, but looking at the graph of e^x will probably convince you of this). This power is called ln(y).

A verbal description would be that ln(y) is the power you need to raise e to to get y. Then e^ln(y) is e raised to the power you need to raise e to to get y. In other words, e^ln(y)=y. When it's stated that way, it's not so mysterious.

 

[rudders93]

Ah ok. Thanks!

 

[annoymage]

e^ln(2) = x

and solve for x.

 

[paulfr]

There is a much deeper and more fundamental reason that
e^ln(2) = 2

Exponentiation and Logarithms are INVERSE FUNCTIONS

Briefly, inverse functions, in layman's terms, are functions which when performed
in succession return you to where you started.
For example; start with x, add 3 to it and then subtract three from it and you are back at x
Addition and Subtraction are Inverse Functions.
Other inverse function pairs are;
1/ Multiplication and Division
2/ Powers and Roots
3/ Trig and Inverse Trig functions
4/ Integration and Differentiation {This is the Fundamental Theorem of Calculus}

Note the reverse of the original expression is ... Ln e^2 is also = 2
This can be simply verified by the Power Rule of Exponents
Ln e^2 = 2 Ln e = 2 x 1 = 2

An important result of this is that whenever you need to solve an
equation, the operation most likely to get you quickly to your answer
is to perform the Inverse Function of the outer operation to both sides.

 

[log10power]

e^ln2 = 2
Apply ln to both sides: ln e^ln2 = ln 2
ln2 ln e = ln 2
Since ln e = 1, so ln2 (1) = ln 2
ln2 = ln2