What base does the author take when taking the log of both sides?

Question

I am learning exponential distribution in ThinkStats2 by Allen Downey.. It says that "if you plot the complementary CDF of a dataset that you think is exponential, you expect to see a function like: $$ y\approx e^{-\lambda x} $$ Then, taking the log of both sides yields:" $$ \log y \approx -\lambda x $$

My question is what base does the author take when taking the log of both sides? I guess that the author takes log of base 10, but it does not explain why we get $-\lambda x$ on the right side of equation. Could someone explain this?

Answer 1

The author takes the natural logarithm on both sides (base $e=2.7182818\cdots $)

Answer 2

In your case author takes base "$e$"

$y\approx e^{-\lambda x} \Leftrightarrow \log_{e} y \approx -\lambda x$

Generally we should guess outgoing from logarithm main property: $$a=b^x \Leftrightarrow \log_{b} a =x$$ or $$a=b^{log_{b} a}$$

Answer 3

The author uses $\log$ base $e$, called the natural log, where $e$ is Euler's number.
$e\approx2.718281828459045$.
In general, mathematicians use $\log x$ to represent the natural $\log$ of $x$
Also, the base of the logarithm will normally be the same as the base of the exponential. This is because
for $A\gt0,\; A\neq1$: $$y=A^x\iff\log_Ay=x$$ In your case, $A=e$, so the base of the $\log$ is $e$.