Understanding the natural log identity: e^(ln x) = x

[Blkmage8]

I've already learned about rules of logs and change of base but I haven't done natural logs yet. I understand why:

ln (e^x) = x since ln (e^x) = x ln e = x

To prove that e^(ln x) = x though I can't figure it out. I know e^x and ln x are inverses though so e^(ln x) is like f(x) = e^x where f(ln x) = e^(ln x) if that means anything... I'm just really having trouble grasping why this is true. I tried plugging in different values of x so I know it works just not why

Also, for a^x = e^(x ln a) I'm confused too. Obviously, I can figure out that it works just not why again. Note that I'm not learning any of this in the classroom so I'm teaching myself.

 

[galactus]

ln(x) is that power to which e must be raised to get x.

Here's a graphical look at it:

Notice that \(\displaystyle e^{x} \;\ and \;\ ln(x)\) are symmetrical about y=x.

 

[Blkmage8]

Thanks! And also, for a^x = e^(x ln a) I'm having troubling understanding why you can just rewrite an exponential term in terms on e and a natural log...

I was thinking that if you have a^x then taking the log of that results in x ln a, which is not equal to a^x. So does it work out that to "negate" the log but still be able to rewrite a^x, you take e and raise in to the x ln a power?

 

[o_O]

\(\displaystyle a^{x} = e^{x ln a}\)

Looking at the right hand side at its exponent, we use the logarithm law:

\(\displaystyle a log_{b} c = log_{b} c^{a}\)

So ...

\(\displaystyle e^{xlna} = e^{lna^{x}} = a^{x} \quad \quad \mbox{Since } e^{ln(y)} = y \mbox{ where } y = a^{x}\)

 

[Deleted member 4993]

Remember the definition:

ln(x) = y means .........................(1)

\(\displaystyle e^y = x\)........................(2)

then using (1)

\(\displaystyle e^{ln(x)} = e^y\)............(3)

using (2) in (3)

\(\displaystyle e^{ln(x)} = e^y \, =\, x\)