How could this be true $n=\log(e^n)$?

Question

I am learning elementary logarithms.

How could this be true $n=\log(e^n)$?

I searched online and couldn't find any info on this, could anyone give me some clue?

Answer 1

The confusion might be that here $\log$ denotes the logarithm to base $e$ the Euler constant. So it is the natural logarithm, often also denoted $\ln$.

Then that $n = \log (e^n)$ is true is just the definition of the logarithm.

Answer 2

For any real $n\geq 0$, we have $$ n=\log_e\left(e^n\right) $$ $$ n=n\log_e\left(e\right) $$ $$ n=n\cdot 1$$ $$ n=n$$

Answer 3

The two function $f(x)=e^x$ and $g(x)=\ln x$ are inverse of each other, so their composition is the identity function: $$gof(x)=\ln(e^x)=x$$