Logarithms · RiskAtoms Academy

By the end you will be able to

1Explain a logarithm as the inverse of a power: it answers "what power gives this number?"
2Use the log rules for products, quotients and powers.
3Use the natural log, ln, and the log button on a calculator.
4Solve (1 + r)ⁿ = k for n, and use the rule of 72 as a quick check.

Why you need it

“At 6% a year, how long until my money doubles?” The unknown is the number of years, and it sits up in the exponent. Logarithms are the tool that brings it down.

L1.5.06 Equity returns and volatility
Log returns, which add up over time.
L1.3.05 Continuous compounding
The natural log, ln, for continuously compounded rates.

The idea

A power, asked backwards

A power answers: “multiply 2 by itself 3 times; what do you get?” (8). A logarithm asks the same thing backwards: “how many times must I multiply 2 to get 8?” (3). It is the inverse of a power.

Power

You know the base and the exponent. Find the result.

2^3 = \,?\quad\Rightarrow\quad 8

Logarithm

You know the base and the result. Find the exponent.

2^{\,?} = 8 \quad\Rightarrow\quad \log_2 8 = 3

Think firstCount the multiplications

How many 2s multiply together to make 32? That is, what is $\log_2 32$ ?
What is $\log_{10} 1000$ ?
What is $\log_{10} 1$ ?

The rules

The definition. One fact, written two ways.

The log rules. Each is a rule of exponents in disguise.

Which log? ln and log₁₀. Calculators have two log buttons. log is the log base 10. ln is the natural log, whose base is a special number called e, about 2.718 (M-A05 is all about it). Finance uses ln far more often. Both follow the same rules.

Solving for time. The reason logs matter in finance.

The rule of 72. A mental shortcut for doubling, and where it comes from.

The rules on one card

\log_b x = y \iff b^y = x \qquad \log_b 1 = 0 \qquad \log_b b = 1

\log(xy) = \log x + \log y \qquad \log\tfrac{x}{y} = \log x - \log y \qquad \log x^p = p \log x

(1 + r)^n = k \;\Rightarrow\; n = \frac{\ln k}{\ln(1 + r)} \qquad \text{doubling: } n \approx \frac{72}{\text{rate in \%}}

How good is the rule of 72? Very good at everyday rates, and worse at high ones:

Rate	Exact years to double	Rule of 72	Error
1%	69.66	72.00	2.34
4%	17.67	18.00	0.33
8%	9.01	9.00	−0.01
12%	6.12	6.00	−0.12
20%	3.80	3.60	−0.20
50%	1.71	1.44	−0.27

Try it

Slide the rate and pick a target. The blue curve is the exact answer from logs. For doubling, the dashed curve is the rule of 72: see where it hugs the exact one and where it drifts away.

Try itHow many years?

Open the logarithms calculator →

Growth per year, r6.0%

Target: multiply the start by k

Solving (1.060)ⁿ = 2 with logs: n = ln 2 ÷ ln 1.060 = 0.6931 ÷ 0.0583

Years to reach ×2

11.90

In whole years12

(1.060)¹²2.012

Rule of 72: 72 ÷ 6.012.00

Rule's error0.10 yrs

exact: ln k ÷ ln(1 + r)rule of 72

Across: the growth rate. Up: the years needed.

Worked examples

1. How long does money take to double at 6% a year?

Step 1 of 3

1
Set up
$1.06^n = 2$

2. How long does 1,000 take to grow to 5,000 at 7% a year?

Step 1 of 3

1
Turn it into a growth factor
5,000 is 5 times 1,000, so solve $1.07^n = 5$ .

3. Log changes add up: a price goes 100 → 110 → 99

Step 1 of 3

1
The log change of each move
$\ln\tfrac{110}{100} = 0.09531 \qquad \ln\tfrac{99}{110} = -0.10536$

Common mistakes

Each of these is tempting. Decide what is wrong before you reveal the fix.

“ln(a + b) = ln a + ln b.”
“ln 2 ÷ ln 1.06 is the same as ln(2 ÷ 1.06).”
“n = ln 2 ÷ log 1.06.”
“At 6%, n = ln 2 ÷ ln(1 + 6).”
“ln 0 = 0”, or taking the log of a negative number.
“At 50% a year, the rule of 72 says money doubles in 1.44 years.”

Reaction

Check what stuck

Question 1 · objective 1

What is log₂ 64?

Question 2 · objective 2

Which of these is always true (for positive a and b)?

Aln(a + b) = ln a + ln bBln(ab) = ln a + ln bCln(ab) = ln a × ln bDln a ÷ ln b = ln(a ÷ b)

Question 3 · objective 4

Exactly how many years does money take to double at 9% a year, compounded yearly? Two decimals.

years

Question 4 · objective 4

Using the rule of 72, roughly how many years does money take to double at 4% a year?

years

Question 5 · objective 4

Solve 1.05ⁿ = 3 for n, to two decimals.

Glossary

Logarithm (log): The power you must raise the base to, to get a number. log₂ 8 = 3 because 2³ = 8.
Natural log (ln): The natural logarithm: a log with base e, a constant close to 2.71828. Written ln.
Log base 10: A log with base 10, written log₁₀ or just log on most calculators.
Inverse: An operation that undoes another. Taking a log undoes raising to a power.
Rule of 72: A shortcut: the years to double at r% a year are roughly 72 ÷ r.
Log change: ln(new ÷ old). Log changes over several periods add up to the log change over the whole time.