Log of e raised to an exponent

Question

I have a textbook which states the following:

$y=e^{(-\lambda x)}$

it then takes the log of both sides and comes up with:

$log \ y = - \lambda x$

Why is the right side what it is? Shouldn't it be:

$log \ y = (- \lambda x)(log \ e)$

?

Answer 1

Some texts and softwares use $\log$ for natural logarithm instead of $\ln$

Answer 2

After reading the comments, I realized that the textbook is referring to $log_e$ and not $log_{10}$ which is what I have thought my entire life.

Answer 3

For logarithms, $$x = a^b \rightarrow \log x = b \log a$$

In this case, you have the natural logarithm ($\log_{e}$, more commonly noted as $\ln$), and the property $$\ln e = 1$$

Hence, $y = e^{-\lambda x} \rightarrow \ln y = \ln e (-\lambda x) \rightarrow \ln y = (1) (-\lambda x) \rightarrow \ln y = -\lambda x$.