Why does e^(ln x) = x?

Answer 1

Let e^ (ln x) = y.

Definition of logarithm.

The logarithm of a number to a given base is the index of the power to which the base must be raised so as to get the number.

The logarithm of the number ‘y’ to a given base “e” is the index of the power (ln x) to which the base ‘e’ must be raised so as to get ‘y ’.

Therefore
ln y = ln x ( both to the base e)

y = x.

e^ (ln x) = y = x.

Answer 2

Let a = e^(lnx)

Take the ln of both sides.

ln a = ln [e^(ln x)]

Use the identity: ln A^b = b ln A, here A = e and b = ln x

ln a = [ln x ] ln e

ln e = 1, (e should be raised to 1 to have a number e.

ln a = ln x

a = x

Answer 3

Exponents and logs are inverse functions. That is, they "undo" each other. If I take 10 to the power 3, the result is 1000. If I take the answer 1000 and then apply the log (base10) of it, I arrive back at 3 ... the power I started with:

10^3 = 1000. Inversely, log 1000 = 3

Now, take the exponential function with base e. The natural log, ln, is the inverse of this function .... it "undoes" exponential functions with base e.

Ex: e^1 = 2.71828... Inversely, ln(2.71828...)=1, more simply written as lne = 1.

Following this same logic, then, let's apply it to your question and then prove
why e^(lnx)=x:

e^(lnx) = x. Inversely, lnx = lnx. Since lnx = lnx is a true statement, then it must be true that e^(lnx) = x.

Answer 4

Because the natural logarithm of x and the function e^x are inverse functions. This fact follows from the fact that ln e^x = x. More concretely, it means that if you put any number into a calculator, then calculate its logarithm, and then raise e to the power (answer of this), you will always get your number back. Example:

ln 2 = .693... (write down all the places your calculator will give).

e^(.693...) = 2

Answer 5

That's the very definition of ln, the natural log. ln x is the exponent to which you have to raise e, to get x.

In other words, that's like asking why (square root of x)² = x
or why x + (-x) = 0.

Answer 6

natural log is the INVERSE of the exponential function of base e. inverse functions, by definition, must have the same input as the output (x in this case).
this is equivalent to saying f(g(x)) = x
so you get back what you put into this function
just in this case, f(x) = e^x and g(x) = ln(x)
hope this makes some sense

Answer 7

This is the definition of ln x - it doesn't need a proof.

Answer 8

ln(x) means that number y , such that

e^y = x.

That is,

y = ln(x) means e^y = x

since y = ln(x), this means e^ (ln(x) = x

Note that this also means that ln(e^x) = x

ln(e^x) is that number y such that e^y = e^x.

That is, y = x.

Kermit < [e-mail] >

Answer 9

when a = ln x, it means that e^a = x
since a = ln x, e^(ln x) = e^a = x

Answer 10

its base is a natural base e