algebra precalculus - Rules of log 2 and simplifying logs

Recall what $\log_2$ means: $2^P =Q$, $\log_2Q=P$

What is $2^{\log N}$? There is a relationship between $2$ and $\log$, so
we should be able to simplify this.

Let $P = 2^{\log N}$. By the definition of $\log_2$ we can write this as
$\log_2P = \log_2N$.This means that $P = N$.

Let $P =2^{ \log N}$

$\log_2P = \log_2N$

$P=N$

$2^{\log N}=N$

I'm confused on the line

By the definition of $\log_2$ we can write this as $\log_2P = \log_2N$

If I multiplied $P = 2^{\log N}$ by $\log_2$, then it'd be $\log_2P =\log_2( 2^{\log N})$

$$\log_2P =\log N \cdot \log_2( 2)$$

$$\log_2P =\log N \cdot 1$$

I'm confused on how we can assume that the base of $\log N$ is two?