natural logarithm
Just as base 10 gives the common logarithm, the base e ≈ 2.718 gives the natural logarithm. It answers: to what power must e be raised to produce this number? Because e is the natural rhythm of continuous growth, its logarithm is the natural tool for measuring how long such growth takes.
We write it as ln(x), a shorthand for log_e(x). So ln(x) = y means precisely e^y = x. It accepts only positive inputs, equals 0 at x = 1, and equals 1 at x = e. The natural log is the exact inverse of the natural exponential function, so ln(e^x) = x and e^(ln x) = x for valid inputs.
The natural log is the workhorse for solving continuous growth and decay problems. When a quantity follows a · e^(kt), taking ln of both sides brings the unknown t out of the exponent and into ordinary arithmetic. This is why half-life and continuous-interest formulas are riddled with ln.
To solve e^(0.1t) = 5, take ln of both sides: 0.1t = ln(5) ≈ 1.609, so t ≈ 16.09. The natural log peels the unknown out of the exponent.
Applying ln converts e^(0.1t) = 5 into a simple linear equation in t.