Free lesson · Math & notation

Powers, square roots, exponentials and logarithms

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Start with the idea

Compounding repeats a growth factor. Roots and logarithms let you solve backwards from ending wealth to the growth rate or elapsed time.

Symbols, units & horizon

V₀, V_T: initial and final positive wealth
T: positive number of years in this example
g: annual decimal compound growth
a, b: positive numbers
ln: natural logarithm, inverse of exp
e: exponential base ≈2.71828
Superscript: power, with 1/T meaning the T-th root

When and why to use this

Use roots for CAGR and logarithms for time-series returns, reversion half-life and implied carry rates.

A power repeats multiplication: 1.05² means 1.05×1.05. A square root reverses squaring for a nonnegative result. An exponential eˣ uses the constant e≈2.71828; the natural logarithm ln reverses that exponential.

V_{T} = V_{0} (1 + g)^{T}, g = (\frac{V_{T}}{V_{0}})^{\frac{1}{T}} - 1, \ln (a b) = \ln a + \ln b

Algebra and arithmetic

Reverse a repeated growth factor

Divide $V_{T} = V_{0} (1 + g)^{T}$ by V₀. Raise both sides to 1/T and subtract 1.
Taking logs instead gives $\ln (\frac{V_{T}}{V_{0}}) = T \ln (1 + g)$ . The identity ln(ab)=ln a+ln b follows because $e^{x} e^{y} = e^{x + y}$ .

Work it by hand

100 growing to 121 over two years has g=√1.21−1=.1. Its log growth per year is ln1.1≈.09531.

The logarithm identity requires positive a and b. It explains why log returns add across time while wealth factors multiply. A calculator supplies logarithms, exponentials and roots; doing a derivation by hand means understanding the sequence of operations rather than memorising a decimal constant.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

from math import exp, log

def compound(start, growth, years):
    return start * (1 + growth)**years

def cagr(start, end, years):
    if min(start, end, years) <= 0:
        raise ValueError("Positive wealth and years required")
    return exp(log(end / start) / years) - 1

print(compound(100, .1, 2), cagr(100, 121, 2))

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