Pure 2: Functions and Logarithms

Pure 2: Functions and Logarithms — CIE A Level Mathematics 9709

1. Logarithms

Definition: log_a(x) = y ⟺ aʸ = x

The logarithm is the inverse of the exponential function.

Laws of Logarithms:

• log_a(xy) = log_a(x) + log_a(y) (product rule)
• log_a(x/y) = log_a(x) − log_a(y) (quotient rule)
• log_a(xⁿ) = n log_a(x) (power rule)
• log_a(1) = 0 (since a⁰ = 1)
• log_a(a) = 1 (since a¹ = a)

Change of base: $log_a x = rac{log_b x}{log_b a}$ (commonly use base 10 or e)

2. The Natural Logarithm and Euler's Number

e = 2.71828... — Euler's number; base of the natural logarithm.

Natural logarithm: ln x = log_e(x)

Key properties: ln(eˣ) = x and e^(ln x) = x.

Differentiation/integration:

• d/dx(ln x) = 1/x
• d/dx(eˣ) = eˣ
• ∫eˣ dx = eˣ + C
• ∫(1/x) dx = ln|x| + C

3. Solving Exponential and Logarithmic Equations

Technique 1: Take logs of both sides

3ˣ = 20 → x ln 3 = ln 20 → x = ln 20 / ln 3 = 2.727

Technique 2: Substitution to form quadratic

2ˣ − 3(2⁻ˣ) = 2 → Multiply by 2ˣ: 2²ˣ − 3 = 2(2ˣ)

Let u = 2ˣ: u² − 2u − 3 = 0 → (u−3)(u+1) = 0 → u = 3 (since u > 0).

2ˣ = 3 → x = ln 3/ln 2 = log₂3 ≈ 1.585

Technique 3: Using ln directly

ln(x+2) + ln(x−1) = ln 10 → ln[(x+2)(x−1)] = ln 10 → (x+2)(x−1) = 10

x² + x − 2 = 10 → x² + x − 12 = 0 → (x+4)(x−3) = 0

x = 3 (x = −4 invalid: ln(negative) undefined). x = 3

4. Logarithmic Graphs — Linearising Data

If y = axⁿ, taking log: log y = log a + n log x (linear: log y vs log x)

If y = aˣ, taking ln: ln y = ln a + x ln b (linear: ln y vs x, for y = abˣ)

Example: Data follows y = kxⁿ. A log-log plot of log y vs log x gives:

• Gradient = n
• y-intercept = log k

This technique converts power/exponential relationships into straight lines for easy analysis.

5. Transformations of y = eˣ and y = ln x

Graph of y = eˣ: Always positive; passes through (0,1); asymptote y = 0; increasing.

Graph of y = ln x: Defined for x > 0 only; passes through (1,0); asymptote x = 0.

Key relationship: y = eˣ and y = ln x are reflections in the line y = x.

Transformations:

• y = e^(2x): horizontal compression factor 2
• y = eˣ + 3: vertical translation 3 up; asymptote now y = 3
• y = ln(x−2): translation 2 right; asymptote x = 2, domain x > 2
• y = −ln x: reflection in x-axis