\( e \) (2.718281828459045…), also known as Euler’s number, is a mathematical constant that shows up when the rate of change in a quantity is proportional to the current value. Some examples from nuclear medicine are radioactive decay and attenuation of gamma photons.
How was e discovered?
Value of \( e \) was discovered by Leonard Euler in 18th century, while trying to solve a problem of compound interest, previously proposed by Jacob Bernoulli. The formula for annual compound interest is \( P \times \left(1 + r/n \right)^n \), where; \( P \) is principal, \( r \) is the interest rate and \( n \) is the number of times interest is compounded per year. For the sake of simplicity, imagine that $1 is invested at an annual interest rate of 100% (=1 as a decimal), so that the formula becomes \( \left(1 + 1/n \right)^n \). What happens if the interest is compounded in shorter and shorter time periods (n is increased)?
\( n \)  \( (1 + 1/n)^n \) 

1 (annually)  2 
2 (biannually)  2.25 
4 (quarterly)  2.44140625 
12 (monthly)  2.61303529… 
52 (weekly)  2.69259695… 
365 (daily)  2.71456748… 
8,760 (hourly)  2.71812669… 
525,600 (minutely)  2.71827924… 
31,536,000 (every second)  2.71828178… 
\( \infty \) (continuously)  \( e \) 
As \( n \) gets closer to infinity, the result converges around 2.71828…
This transcendental number, called \( e \), represents the base rate of change, shared by all exponentially changing processes and can be defined as;
\[ e = \lim\limits_{n\to \infty} \left(1 + \dfrac{1}{n} \right)^n \]
Apply binomial theorem on this definition and you’ll get an even fancier one;
\[ e = \sum\limits_{n=0}^\infty \dfrac{1}{n!} = \dfrac{1}{0!}+\dfrac{1}{1!}+\dfrac{1}{2!}+\cdots \]
Meaning of \( e^x \) and \( \ln (x) \)
 \( e^x \) gives the amount of growth with a certain rate of change, after a certain amount of time, where \(x = rate \cdot time\).
 \( \ln (x) \) gives the amount of time (and rate) needed to reach a certain level of growth (\( x \)).
Examples from nuclear medicine

Radioactive decay
\( N_t = N_0 e^{\lambda t} \)
where;
\( N_t \) is the quantity at time \( t \),
\( N_0 \) is the initial quantity,
\( \lambda \) is exponential decay constant (\( 0.693 \, / \, t_{1/2} \)). 
Attenuation of gamma photons
\( I = I_0 e^{\mu x} \)
where;
\( I \) is the intensity of photons after attenuation,
\( I_0 \) is the initial intensity of photons,
\( \mu \) is the linear attenuation coefficient of medium,
\( x \) is the distance travelled in medium.
Some cool properties of e
 At any point, the slope of \( e^x \) is equal to the value of \( e^x \).
 The area under curve \( e^x \), from \( \infty \) up to any x, is also equal to \( e^x \).
 The global maximum of \( \sqrt[x]{x} \) occurs at \( x = e \).
 Probably the most beautiful equation in the universe: \( e^{i\pi} + 1 = 0 \).