Exponents & Logarithms

Index laws, special cases, the number e, and exponential growth & decay. ✎

① Index Laws ✦ booklet

m+n

same base → add exponents

same base → subtract exponents

power of power → multiply exponents

power splits over a product

power splits over a quotient

② Special Indices

a 0 = 1

any base (a ≠ 0)

a −n = 1 / a n

negative → reciprocal

a 1/n = n √a

fraction → root

root then power

Exam tip!

③ Graph of y = a x

y = 0

(asymptote)

y = aˣ (a>1)

y-intercept always (0, 1) — any base raised to 0 is 1.

Horizontal asymptote: y = 0 (never crosses the x-axis).

0 < a < 1 → decay (mirror image, falling left to right).

④ Graph of y = ab x + c

y=0

y=c

y=bˣ

y=abˣ+c

Asymptote shifts from y = 0 to y = c — adding c lifts the whole curve.

y-intercept: set x = 0 → y = a·b⁰ + c = a + c.

c can be negative — the asymptote shifts below the x-axis.

a < 0 flips the curve — it then approaches y = c from below.

⑤ The Number e

e ≈ 2.71828…

e is irrational — like π, it goes on forever.

y = eˣ is the natural exponential function — most important in maths.

Same graph shape as y = aˣ: passes through (0, 1), asymptote y = 0.

Its inverse is the natural log ln: e ln x = x and ln(eˣ) = x.

e 0 = 1

e 1 = e

e −1 ≈ 0.37

⑥ Growth & Decay Models

Discrete model — multiplied each period:

y = A · b t

Continuous model — using e:

y = A · e kt

A = initial value (when t = 0).

b > 1 → growth · 0 < b < 1 → decay (b = 1+r for rate r).

k > 0 → growth · k < 0 → decay.

Convert: b = e k → k = ln b.

★ Worked example ✎

A colony starts with 200 bacteria and doubles every 3 hours.

After 9 h:

1 600 bacteria

Initial amount when t=0: N = 200 · 2⁰ = 200 ✓

When does N = 5 000? → Switch to Logarithms tab to solve this! ✦

Logarithms

Definition, laws, the natural log, graphs & solving exponential equations. ✎

① What is a Logarithm?

The inverse of exponentiation. If a y = x, then:

log a (x) = y ⟺ a y = x

" log base a of x equals y" → "a raised to y gives x"

Quick examples:

log 2 (8) = 3 because 2 3 = 8

log 10 (100) = 2 because 10 2 = 100

log 3 (1) = 0 because 3 0 = 1

② Laws of Logarithms ✦ booklet

Product rule

Quotient rule

Power rule

③ Special Values & Change of Base

Always-true log values:

log a (1) = 0

log a (a) = 1

log a (a x ) = x

a log a (x) = x

Change of base ✦ booklet

ln x

ln b

log x

log b

Useful when your calculator only has ln or log₁₀ buttons.

④ Natural Logarithm ln

ln x = log e (x)

e x and ln x are inverses: they undo each other.

ln(e x ) = x and e ln x = x

ln(1) = 0

ln(e) = 1

⑤ eˣ and ln x — Mirror Images

y=x

y=eˣ

y=ln x

y = ln x is the reflection of y = eˣ across the line y = x.

eˣ has asymptote y = 0 · passes through (0, 1).

ln x has asymptote x = 0 · passes through (1, 0).

⑥ Solving Exponential Equations

Method 1 — Equal bases

x+1

x + 1 = 3 → x = 2

Method 2 — Take ln of both sides

Method 3 — For e-based equations

2x

x = ln 5 / 2 ≈ 0.805

★ Worked example ✎ — finishing the bacteria problem

N = 200 · 2 t/3. When does N = 5 000?

200 · 2 t/3 = 5 000 → 2 t/3 = 25

t = 3 · ln 25 / ln 2 ≈ 3 × 4.644 ≈ 13.9 hours

Always write your equation first, then solve step by step — show the log step explicitly for full marks.

made for revision · IB AA SL · keep it by your desk ✦