natural exponential function
Among all exponential functions b^x, one base is so convenient that mathematicians treat it as the default. That base is the number e ≈ 2.718, and the function e^x is called the natural exponential function, often written exp(x). It is the exponential that describes smooth, continuous growth and decay in physics, biology, and finance.
Like every exponential with base greater than 1, e^x is always positive, passes through (0, 1), rises as x increases, and has the x-axis as a horizontal asymptote on the left. What makes it special is a calculus property: e^x is its own derivative, meaning its steepness at any point equals its value there. No other base does this exactly.
In practice, any growth or decay model written as a · b^t can be rewritten in the natural form a · e^(kt), where k is a single rate constant: k > 0 for growth, k < 0 for decay. This unified form is why scientists overwhelmingly prefer base e over messy bases like 1.07 or 0.83.
A continuous 6% growth is e^(0.06t). At t = 1, e^0.06 ≈ 1.0618, so the effective yearly factor is about 6.18%, slightly more than 6% because the growth is compounded continuously.
Continuous compounding turns a 6% rate into a 6.18% effective factor.