Logarithms Guide

For any base b (b > 0, b ≠ 1):

log_b 1 = 0

This is because b⁰ = 1 for any base b.

For any base b (b > 0, b ≠ 1):

log_b b = 1

This is because b¹ = b.

Logarithms and exponents are inverse operations:

log_b (bˣ) = x

b^{log_b x} = x

• log_b 0 is undefined

• log_b (-x) is undefined for real numbers

• The base b must be positive and not equal to 1

Base 10 logarithm, written as log x (without the base)

log x = log₁₀ x

Uses: Scientific notation, pH scale, Richter scale

Examples:

log 100 = 2 (since 10² = 100)

log 0.01 = -2 (since 10⁻² = 0.01)

Base e logarithm, written as ln x

ln x = log_e x

where e ≈ 2.71828 (Euler's number)

Uses: Calculus, exponential growth/decay, compound interest

Examples:

ln e = 1 (since e¹ = e)

ln 1 = 0 (since e⁰ = 1)

You can convert between common and natural logs:

log x = ln x / ln 10

ln x = log x / log e

Since ln 10 ≈ 2.3026 and log e ≈ 0.4343

Binary Logarithm: Base 2, written as lb x or log₂ x

Used in computer science and information theory

Arbitrary Bases: Any positive number except 1 can be a base

log_b x represents the exponent to which b must be raised to get x

The logarithm of a product is the sum of the logarithms:

log_b (mn) = log_b m + log_b n

Example:

log₂ (8 × 4) = log₂ 8 + log₂ 4 = 3 + 2 = 5

Check: 2⁵ = 32, and 8 × 4 = 32

The logarithm of a quotient is the difference of the logarithms:

log_b (m/n) = log_b m - log_b n

Example:

log₃ (27/9) = log₃ 27 - log₃ 9 = 3 - 2 = 1

Check: 3¹ = 3, and 27/9 = 3

The logarithm of a power is the exponent times the logarithm of the base:

log_b (mⁿ) = n · log_b m

Example:

log₂ (4³) = 3 · log₂ 4 = 3 × 2 = 6

Check: 2⁶ = 64, and 4³ = 64

Logarithms can be converted to any base using:

log_b a = log_c a / log_c b

Example:

log₈ 64 = log₂ 64 / log₂ 8 = 6 / 3 = 2

Check: 8² = 64

Equations of the form log_b x = c

Solution: Convert to exponential form

If log_b x = c, then x = b^c

Example:

log₃ x = 4 → x = 3⁴ = 81

If log_b m = log_b n, then m = n

Example:

log₂ (x+3) = log₂ (2x-1)

x+3 = 2x-1 → x = 4

Check: log₂ 7 = log₂ 7 ✓

Apply log rules to simplify before solving

Example:

log x + log (x-3) = 1

log [x(x-3)] = 1 → x(x-3) = 10

x² - 3x - 10 = 0 → (x-5)(x+2) = 0

x = 5 (x = -2 is extraneous)

• Always check for extraneous solutions

• The argument of a log must be positive

• Watch for domain restrictions

• Some equations may require numerical methods

log_b a = log a / log b

Example: Evaluate log₅ 25

log₅ 25 = log 25 / log 5

= 1.3979 / 0.6990 ≈ 2

Check: 5² = 25 ✓

log_b a = ln a / ln b

Example: Evaluate log₃ 81

log₃ 81 = ln 81 / ln 3

= 4.3944 / 1.0986 ≈ 4

Check: 3⁴ = 81 ✓

Changing the base of a logarithm stretches or compresses the graph vertically.

log_b x = (1 / log_a b) · log_a x

The graphs have the same general shape but different vertical scaling.

• log_b a = 1 / log_a b

• log_bⁿ a = (1/n) · log_b a

• log_b aⁿ = n · log_b a (power rule)

Compound Interest: A = P(1 + r/n)^nt

Solving for t: t = log(A/P) / [n · log(1 + r/n)]

Economic Growth: Measuring GDP growth rates

Stock Market: Analyzing returns over time

pH Scale: pH = -log[H⁺]

Sound Intensity: Decibel scale: dB = 10·log(I/I₀)

Earthquake Magnitude: Richter scale

Radioactive Decay: Half-life calculations

Algorithm Analysis: Time complexity (O(log n))

Information Theory: Measuring information content

Data Compression: Logarithmic scaling of data

Binary Search: Requires O(log n) time

Logarithmic Scales: For data spanning multiple orders of magnitude

Log Transformations: To linearize exponential relationships

Statistical Modeling: Log-normal distributions

Machine Learning: Logistic regression