F: Exponentials and logarithms flashcards

Topic 7: F: Exponentials and logarithms

Definition of a logarithm

For a > 0 and x > 0: y = logₐ x ⇔ aʸ = x. The log is the power. ln x = logₑ x.

Laws of logarithms

logₐ x + logₐ y = logₐ(xy) logₐ x − logₐ y = logₐ(x/y) k logₐ x = logₐ(xᵏ)

Key features of y = eˣ and y = ln x

y = eˣ: through (0, 1), asymptote y = 0, range y > 0. y = ln x: through (1, 0), asymptote x = 0, domain x > 0. They are reflections of each other in y = x.

Gradient of y = e^(kx)

dy/dx = k e^(kx): the gradient is proportional to the value of the function.

Solve e^(2x) = 5 and 3ˣ = 20.

e^(2x) = 5: 2x = ln 5, x = ½ ln 5 (= 0.805). 3ˣ = 20: x = ln 20/ln 3 = 2.73 (3 s.f.).

Which graphs make y = axⁿ and y = kbˣ straight lines?

y = axⁿ: log y against log x (gradient n, intercept log a). y = kbˣ: log y against x (gradient log b, intercept log k).

The exponential model y = Ae^(kt)

A is the initial value (t = 0). k > 0 gives growth, k < 0 gives decay. y = c + Ae^(kx) has asymptote y = c.

A quantity decays as N = N₀e^(−kt). Find the half-life.

½N₀ = N₀e^(−kt), so e^(−kt) = ½ and t = ln 2/k.

Limitations of exponential growth models

Real quantities cannot grow without limit (resources, space), so the model usually fails for large t. Refine it, for example with a limiting value.