Maths⏱ 5 min read
What Is a Logarithm and How Do You Calculate One?
Logarithms appear in everything from decibels to earthquakes to compound interest. Here's what they actually mean, how to calculate them, and the key rules that make them useful.
A logarithm answers the question: "What power do I need to raise this base to, in order to get this number?" They're inverse operations of exponentiation — the same relationship that subtraction has to addition.
The Definition
log_b(x) = y means b^y = x
"The log base b of x equals y" means "b raised to power y equals x"
log_10(1000) = 3 because 10^3 = 1000
log_10(100) = 2 because 10^2 = 100
log_10(10) = 1 because 10^1 = 10
log_10(1) = 0 because 10^0 = 1
log_10(0.1) = -1 because 10^-1 = 0.1
The Three Common Bases
Base 10 (common log):
log(x) or log_10(x)
Used in: decibels, pH, earthquake magnitude (Richter scale)
Base e (natural log, e = 2.71828...):
ln(x) or log_e(x)
Used in: continuous compounding, population growth, calculus
Base 2 (binary log):
log_2(x)
Used in: information theory, computer science (bits)
Key Log Rules
Product rule: log(a x b) = log(a) + log(b)
Quotient rule: log(a / b) = log(a) - log(b)
Power rule: log(a^n) = n x log(a)
Change of base: log_b(x) = log(x) / log(b)
Examples:
log(200) = log(2 x 100) = log(2) + log(100) = 0.301 + 2 = 2.301
log(50) = log(100/2) = log(100) - log(2) = 2 - 0.301 = 1.699
log(10^4.5) = 4.5 x log(10) = 4.5
Change of base: log_2(32) = log(32)/log(2) = 1.505/0.301 = 5
Check: 2^5 = 32 ✓
Real-World Applications
Decibels (dB): Sound intensity is measured on a logarithmic scale because human hearing perceives loudness logarithmically. A 10 dB increase sounds "twice as loud" but represents 10 times the acoustic power.
dB = 10 x log_10(I / I_0)
Where I_0 = threshold of hearing (10^-12 W/m2)
Whisper: 30 dB = 10^-9 W/m2
Conversation: 60 dB = 10^-6 W/m2 (1000x more power than whisper)
Rock concert: 110 dB = 10^-1 W/m2
Richter scale: Each whole number increase represents 10x more amplitude and approximately 31.6x more energy released.
Compound interest: Logarithms solve for time:
At what rate does £1,000 double in 10 years?
2 = (1+r)^10
log(2) = 10 x log(1+r)
log(1+r) = 0.3010 / 10 = 0.03010
1+r = 10^0.03010 = 1.0718
r = 7.18% per year
Log vs Ln on a Calculator
log button: base 10 (common logarithm)
ln button: base e (natural logarithm)
Converting between them:
ln(x) = log(x) / log(e) = log(x) / 0.4343
ln(x) = 2.3026 x log(x)
log(x) = ln(x) / ln(10) = ln(x) / 2.3026