Logarithms Without Fear: What They Mean and Where They Appear

The one-sentence idea

A logarithm answers the question: how many times do I multiply this base by itself to get that number?

2 × 2 × 2 = 8, so the base-2 logarithm of 8 is 3. Written: log₂(8) = 3.
10 × 10 × 10 = 1,000, so log₁₀(1,000) = 3.
5 × 5 = 25, so log₅(25) = 2.

In other words, a logarithm is the exponent. The statements 10^3 = 1,000 and log₁₀(1,000) = 3 say exactly the same thing in two directions. Powers ask "what do I get?"; logarithms ask "what power did I need?".

Beyond whole numbers

Logarithms also work for numbers that are not neat powers. Since 100 = 10^2 and 1,000 = 10^3, the base-10 logarithm of 500 must lie between 2 and 3. A calculator gives log(500) ≈ 2.699, which means 10^2.699 ≈ 500.

log(1) = 0 for any base, because any base raised to the power 0 is 1.
Numbers between 0 and 1 have negative logs. log(0.01) = −2, because 10^−2 = 1/100.
log(0) and logs of negative numbers are undefined. No power of a positive base gives zero or a negative number, so the calculator shows an error.

A handy use: the number of digits in a positive whole number is the whole-number part of its base-10 log, plus one. 2^10 = 1,024 has log ≈ 3.01, so it has 4 digits. 2^64 has log ≈ 64 × 0.30103 ≈ 19.27, so it has 20 digits; indeed 2^64 = 18,446,744,073,709,551,616.

log versus ln

Calculators usually offer two logarithm keys.

Key	Base	Common in	Example
log	10	Scales such as decibels and pH, counting digits	log(1,000) = 3
ln	e ≈ 2.718281828	Growth and decay, calculus, physics	ln(e) = 1

The number e appears naturally whenever something grows in proportion to its current size, which is why ln is called the natural logarithm. Some books also use log₂, the binary logarithm, which counts doublings: log₂(256) = 8, so 8 bits can store 256 different values.

A few values worth remembering: log(2) ≈ 0.30103, log(3) ≈ 0.47712, log(5) ≈ 0.69897, and ln(2) ≈ 0.693147. Notice that log(2) + log(5) = 1, because 2 × 5 = 10.

The three log rules

Because logarithms are exponents, they follow the rules of exponents, turning multiplication into addition.

Product rule: log(a × b) = log(a) + log(b). Example: log(2) + log(5) = log(10) = 1.
Quotient rule: log(a ÷ b) = log(a) − log(b). Example: log(1,000 ÷ 10) = 3 − 1 = 2.
Power rule: log(a^n) = n × log(a). Example: log(10^5) = 5, and log(√10) = 0.5 × log(10) = 0.5.

Before electronic calculators, these rules let people multiply large numbers by adding their logarithms, using printed log tables or a slide rule. Logarithms were published by John Napier in 1614 for exactly this purpose.

Watch out for rules that look right but are not. log(a + b) is not log(a) + log(b): log(1 + 1) = 0.301, but log(1) + log(1) = 0. And log(a) ÷ log(b) is not log(a ÷ b).

Change of base: any log on any calculator

Most calculators only have log and ln. To get a logarithm in another base, divide:

log_b(x) = log(x) ÷ log(b) = ln(x) ÷ ln(b)

log₂(100) = log(100) ÷ log(2) = 2 ÷ 0.30103 ≈ 6.644. Check: 2^6.644 ≈ 100.
log₃(81) = ln(81) ÷ ln(3) = 4, since 3^4 = 81.
log₅(30) = log(30) ÷ log(5) ≈ 1.47712 ÷ 0.69897 ≈ 2.113.

On the scientific calculator, type log(100)÷log(2). The brackets around each argument matter.

Solving for an unknown exponent

Logs are the tool for questions such as "2 to what power is 50?". Take the log of both sides of 2^x = 50 and use the power rule: x × log(2) = log(50), so x = log(50) ÷ log(2) ≈ 5.644. Check: 2^5 = 32 and 2^6 = 64, so an answer between 5 and 6 makes sense.

Doubling time for steady growth

When a quantity grows by r% each period, it is multiplied by (1 + r/100) each time. To find how many periods it takes to double, solve (1 + r/100)^n = 2:

n = ln(2) ÷ ln(1 + r/100)

Growth per period	Exact doubling time	"Rule of 72" estimate (72 ÷ r)
3%	23.45 periods	24
5%	14.21 periods	14.4
7%	10.24 periods	10.29

The rule of 72 is a mental shortcut that is close for small growth rates. This is pure arithmetic about steady growth, such as a colony of cells or a website's visitors growing at a fixed rate; real-world quantities rarely grow at a perfectly steady rate. The percent change guide shows how repeated growth builds up.

Decay works the same way in reverse. A radioactive sample with a half-life of 5 years falls to 1/16 of its amount after 20 years (four halvings). To find when it reaches 10%, solve (1/2)^(t/5) = 0.1: t = 5 × log(10) ÷ log(2) ≈ 16.6 years.

Logarithmic scales in the real world

When a quantity ranges over many powers of ten, scientists often measure it on a log scale, where each step means "multiply by a fixed amount" rather than "add a fixed amount".

Earthquake magnitude. On the magnitude scales used by seismologists, each whole step up corresponds to about 10 times larger ground motion as recorded by instruments and about 10^1.5 ≈ 31.6 times more energy released. A magnitude 7 earthquake therefore releases about 31.6 × 31.6 ≈ 1,000 times the energy of a magnitude 5.
Decibels. Sound intensity level is 10 × log(I ÷ I₀), where I₀ is a reference intensity. Every +10 dB means 10 times the intensity, so 90 dB is 1,000 times the intensity of 60 dB. Doubling the intensity adds 10 × log(2) ≈ 3 dB, which is why two equal 70 dB sources together measure about 73 dB, not 140 dB.
pH. pH is the negative base-10 log of the hydrogen ion concentration. A drop of one pH unit means 10 times the concentration, so a solution at pH 3 has 100 times the hydrogen ion concentration of one at pH 5.
Log-scale charts. On a chart with a logarithmic axis, equal distances mean equal ratios. Steady percentage growth appears as a straight line, which makes changes in the growth rate easy to spot.

Common mistakes

Mixing up log and ln. log(100) = 2, but ln(100) ≈ 4.605. Check which one a formula expects.
Splitting a sum. log(a + b) cannot be broken into separate logs.
Taking the log of zero or a negative number. It has no real answer.
Reading log scales as linear. Magnitude 6 is not "20% more" than magnitude 5; it is about 31.6 times the energy.
Forgetting brackets in change of base. Type log(100)÷log(2), not log 100÷2.

FAQ

What is the difference between log and ln?
Only the base. log uses 10 and ln uses e. Any result in one can be converted to the other: ln(x) = log(x) ÷ log(e) ≈ 2.302585 × log(x).

Why is log(1) zero?
Because any positive base raised to the power 0 equals 1, so you multiply the base zero times.

Do I need logarithms in everyday life?
You use them indirectly whenever you read a decibel level, a pH value or an earthquake magnitude, or estimate how long steady growth takes to double. For a quick refresher on powers and E notation, see the scientific calculator guide.

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