Exponential and logarithmic functions

Powers are left far behind by exponentials. At x = 10³, x¹⁰⁰ is 10³⁰⁰, while 10ˣ is 10¹⁰⁰⁰.

For a base b > 1, y = bˣ has domain R and range the positive reals, passes through (0, 1), always increases, and hugs the x-axis for large negative x without touching it.

Base 10 gives the common exponential. The number e, between 2 and 3, is the sum 1 + 1/1! + 1/2! + …; base e gives the natural exponential eˣ.

log_b a = x means bˣ = a. So log₂ 8 = 3, log₁₀ 10000 = 4, log₅ 625 = 4 and log₂₅ 625 = 2. The natural log log x uses base e.

For b > 1, log_b x is defined only for x > 0, increases, is negative for 0 < x < 1, zero at 1, and tends to −∞ as x → 0⁺. Its graph is the mirror image of bˣ in the line y = x.

Rules: log_a p = log_b p / log_b a, log(pq) = log p + log q, log(pⁿ) = n log p, log(p/q) = log p − log q. So x = e^(log x) holds only for x > 0.

Key derivatives: d/dx(eˣ) = eˣ, and d/dx(log x) = 1/x for x > 0. Both are continuous on their domains.

With the chain rule: e⁻ˣ gives −e⁻ˣ; sin(log x) gives cos(log x)/x; e^(cos x) gives −sin x · e^(cos x); cos⁻¹(eˣ) gives −eˣ/√(1 − e²ˣ).