What Is a Logarithm?
A logarithm is the inverse operation of exponentiation. Just as subtraction undoes addition and division undoes multiplication, logarithms undo raising a number to a power. If bˣ = y, then log base b of y equals x. Logarithms make complex calculations simpler and are used across science, engineering, finance, and computer science.
Common Types of Logarithms
- Common logarithm (log): base 10, written as log₁₀ or just log. Widely used in science and engineering.
- Natural logarithm (ln): base e (≈2.71828), written as ln. Fundamental in calculus, exponential growth/decay, continuous compounding.
- Binary logarithm (log₂): base 2. Used in computer science, information theory, and music theory.
- Arbitrary base: any positive base b ≠ 1 can be used with the change of base formula.
Change of Base Formula
To find a logarithm with any base, use the change of base formula:log_b(x) = log_c(x) / log_c(b) = ln(x) / ln(b)This lets you calculate any base using either natural log or common log. For example, log₂(10) = ln(10) / ln(2) ≈ 2.3026 / 0.6931 ≈ 3.3219.
Logarithm Rules and Properties
- Product Rule: log_b(x · y) = log_b(x) + log_b(y)
- Quotient Rule: log_b(x / y) = log_b(x) − log_b(y)
- Power Rule: log_b(xⁿ) = n · log_b(x)
- Identity: log_b(b) = 1 and log_b(1) = 0
- Reciprocal: log_b(1/x) = −log_b(x)
Worked Examples
Example 1: What is log₁₀(1000)? Since 10³ = 1000, log₁₀(1000) = 3.
Example 2: What is ln(e⁵)? Since e⁵ = e⁵, ln(e⁵) = 5.
Example 3: Simplify log₂(8) + log₂(4). Using the product rule: log₂(8 × 4) = log₂(32) = 5.
Example 4: Find log₃(81). 3⁴ = 81, so log₃(81) = 4.
Real-World Applications
Logarithms appear in many fields. In acoustics, the decibel (dB) scale is logarithmic. In seismology, the Richter scale measures earthquake magnitude on a log base-10 scale. In chemistry, pH is the negative log of hydrogen ion concentration. In finance, logarithms calculate compound interest and doubling times. In computer science, binary logarithms measure algorithm complexity and data storage. In statistics, log transformations are used to handle skewed data and multiplicative relationships.
Exponential vs. Logarithmic Growth
Exponential functions grow faster and faster (e.g., compound interest, viral spread). Logarithmic functions grow slower and slower — they describe things that start quickly and then level off (e.g., learning curves, information gain). The two are inverses: the log of an exponential (or vice versa) brings you back to the original value.
Frequently Asked Questions
Can you take the log of zero or a negative number?
No, in real numbers, you can only take the logarithm of a positive number. There is no real power to which you can raise a positive base and get zero or a negative result.
What is the relationship between ln and log?
log₁₀(x) = ln(x) / ln(10) ≈ ln(x) / 2.3026. Conversely, ln(x) ≈ 2.3026 × log₁₀(x).
How are logarithms used in finance?
The Rule of 72 (a quick way to estimate doubling time) is based on logarithms: time to double ≈ 69.3 / interest rate for continuous compounding (72 is used as a convenient approximation). Log returns also play an important role in modern portfolio theory.