Section 12

Exponentials and logarithms

fundamentalviz

f (x) = a^{x}, a > 0, a \neq = 1

Basic model of exponential growth and decay.

fundamentalviz

lo g_{b} x = y ⟺ b^{y} = x

Equivalent form

b > 0, b \neq = 1, x > 0

The logarithm is the inverse of exponentiation.

fundamental

lo g_{b} (x y) = lo g_{b} x + lo g_{b} y

Converts products into sums.

fundamental

lo g_{b} (x / y) = lo g_{b} x - lo g_{b} y

Converts quotients into differences.

fundamental

lo g_{b} (x^{r}) = r lo g_{b} x

Converts exponents into factors.

fundamental

lo g_{b} x = \frac{lo g _{a} x}{lo g _{a} b}

Expresses a logarithm in any valid base.

fundamentalviz

P (t) = P_{0} e^{k t}

Positive k produces growth; negative k produces decay.