Looking at e

[Jason76]

Another way to see \(\displaystyle e\) or Euler's Constant:

\(\displaystyle log_e(1) = 0\) which is understood as \(\displaystyle \ln(1) = 0\) - Understood verbally as: "The natural \(\displaystyle log\) of \(\displaystyle 1 = 0\)".

Therefore:

\(\displaystyle e^{0} = 1\)

Is this reasoning ok?

 

[JeffM]

Another way to see \(\displaystyle e\) or Euler's Constant:

\(\displaystyle log_e(1) = 0\) which is understood as \(\displaystyle \ln(1) = 0\) - Understood verbally as: "The natural \(\displaystyle log\) of \(\displaystyle 1 = 0\)".

Therefore:

\(\displaystyle e^{0} = 1\)

Is this reasoning ok?

\(\displaystyle For\ any\ real\ number\ b > 0\ and\ b \ne 0,\ log_b(1) = 0.\)

You are still trying to memorize formulas instead of understanding principles.

\(\displaystyle Given\ b > 0 < a\ and\ b \ne 1,\ log_b(a) = c \iff a = b^c.\) This is a definition.

But \(\displaystyle b^0 = 1.\)

So \(\displaystyle log_b(1) = 0.\)

 

[daon2]

\(\displaystyle e\) can also be defined as (the unique) number such that

\(\displaystyle \dfrac{e^x-1}{x} \to 1\) as \(\displaystyle x\to 0\)

 

[HallsofIvy]

Another way to see \(\displaystyle e\) or Euler's Constant:

\(\displaystyle log_e(1) = 0\) which is understood as \(\displaystyle \ln(1) = 0\) - Understood verbally as: "The natural \(\displaystyle log\) of \(\displaystyle 1 = 0\)".

Therefore:

\(\displaystyle e^{0} = 1\)

Is this reasoning ok?

That's true but it says nothing about "e". \(\displaystyle a^0= 1\) for any positive number, a.