Can someone explain to me why ln(e^x) was x please?

[Kaiya]

Although I understand why ln and e cancel each other, I am confused because of the exponent

[garypettie]

Hi Kaiya,
You can think of logarithms a bit like the undo function for exponents.

Let’s take the standard form of log_b(b^x) = x and throw some actual numbers in there to show why it works.

Let’s say we have 10^x = 100 and want to find the value of x.
Using logarithms we can say that log_10(100) = 2.
Therefore, x = 2.
We can confirm that this is true because 10^2 = 100.

Likewise, if you had 2^x = 64
you could find x with log_2(64) = 6.
Again we could confirm this because 2^6 = 64.

Now, when whenever you see logarithms written without specifying a base, you’ll see one of two forms.
Log is just Log_10, also known as the common log.
Ln is the same as Log_e, otherwise known as the natural log.

Remember that e is really just a number and if you convert it to decimal it’s around 2.718 (3dp).
However, since it’s a particularly useful number, it’s just been given a special name and symbol - a bit like π (pi) or φ (phi / the “golden” ratio).

I hope that helps clear things up but if you have any other questions, please let me know.

[Kaiya]

@garypettie

log_b(b^x)

Thanks, so is it x because the standard form ‘log_b(b^x)’ is all canceled out making the right side = x the solution and not because of the exponent x?

[garypettie]

Yep. log_b(b^x) cancels everything out except for the x.

In practical use, you generally have something like b^x = ?, so you can then use log_b(?) to find the value of x.
The basic formula just wraps all of that up into a single line to make it easier to work with algebraically.
We cover algebraic manipulation in a lot more detail in the next section of the course.