Apply e and ln to model continuous growth, decay, and compound interest.
The natural base e ≈ 2.71828… arises naturally in continuous growth processes. The natural exponential function eˣ is its own derivative, making it indispensable in calculus. The natural logarithm ln(x) = log_e(x) is its inverse.
$1000 invested at 5% annual interest compounded continuously for 10 years: A = 1000e^(0.05 · 10) = 1000e^0.5 ≈ 1000(1.6487) ≈ $1648.72.
Half-life of 14 hours. Find the decay constant k: 0.5 = e^(14k) → ln(0.5) = 14k → k = −ln(2)/14 ≈ −0.0495. After 42 hours: A = A₀e^(−0.0495·42) ≈ 0.125A₀ = A₀/8.
Remember This!
Half-life formula: t_{1/2} = ln(2)/k. Doubling time (for growth): t_d = ln(2)/r. These shortcuts come directly from solving A(t) = 2A₀ or A(t) = A₀/2.
A bacterial culture grows from 100 to 3000 in 3 hours using A(t) = A₀e^(rt). What is the approximate growth rate r?