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Logarithms Explained

A logarithm answers the question: what exponent do I need to raise this base to, to get this number? Logarithms are the inverse of exponentiation — and they appear in pH scales, earthquake magnitudes, sound levels, compound interest, and throughout information theory.

In this lesson

  1. What a logarithm is
  2. Log notation and the three forms
  3. Logarithm rules
  4. Natural log and e
  5. Real-world applications

1 What a Logarithm Is

log_b(x) = y means b^y = x. Read as "log base b of x equals y." The logarithm asks: to what power must I raise b to get x?

Example: log₂(8) = 3 because 2³ = 8. log₁₀(1000) = 3 because 10³ = 1000. log₅(25) = 2 because 5² = 25.

Logarithm as inverse

Exponentiation: base^exponent = result. Logarithm: log_base(result) = exponent. They're inverse operations. Just as subtraction undoes addition, logarithms undo exponentiation.

2 Log Notation and the Three Forms

Every logarithmic equation has an equivalent exponential form:

Converting Between Forms
Convert log₃(81) = 4 to exponential form and verify
1Logarithmic form: log₃(81) = 4
2Exponential form: 3⁴ = 81
3Verify: 3 × 3 × 3 × 3 = 81 ✓
Answer: 3⁴ = 81

Common log: log(x) without a base means log₁₀(x). Used in pH, decibels, earthquake scales.
Natural log: ln(x) means log_e(x) where e ≈ 2.71828. Used in calculus, exponential growth, information theory.

3 Logarithm Rules

The core log rules
1Product rule: log(ab) = log(a) + log(b)
2Quotient rule: log(a/b) = log(a) − log(b)
3Power rule: log(aⁿ) = n·log(a)
4Change of base: log_b(x) = log(x)/log(b) = ln(x)/ln(b)
5Identity: log_b(b) = 1 and log_b(1) = 0
Using Log Rules
Simplify log₂(32) + log₂(4)
1Product rule: log₂(32 × 4) = log₂(128)
2128 = 2⁷
3log₂(2⁷) = 7
Answer: 7

4 Natural Log and e

e ≈ 2.71828 is Euler's number — the base of natural logarithms. It arises naturally (hence the name) in continuous growth and decay problems, compound interest at continuous compounding, and is the unique number where d/dx(eˣ) = eˣ.

ln(x) = log_e(x). Properties: ln(e) = 1, ln(1) = 0, ln(eˣ) = x, e^(ln x) = x.

The natural log is essential in calculus: ∫(1/x)dx = ln|x| + C. This is why e appears throughout advanced mathematics — it makes calculus clean.

5 Real-World Applications

pH scale: pH = −log₁₀[H⁺]. The negative log compresses the enormous range of hydrogen ion concentrations (from 1 to 0.0000001 mol/L) into the familiar 0-14 scale. Each unit is a factor of 10.

Earthquakes: the Richter scale is logarithmic. A magnitude 7 earthquake is 10× more powerful than magnitude 6, and 100× more than magnitude 5.

Decibels: sound intensity in dB = 10·log₁₀(I/I₀). Human hearing spans a factor of 10 trillion in intensity — logarithms compress this to 0-130 dB.

Information theory: entropy (information content) = −Σ p·log(p). Shannon's information entropy underlies data compression, cryptography, and machine learning.

Practice Problems

Evaluate log₂(64)
2 to what power = 64? 2⁶ = 64. Answer: 6
Convert to exponential form: log₅(125) = 3
5³ = 125
Simplify: log(100) + log(10) using log rules
log(100×10) = log(1000) = log(10³) = 3
Use change of base to find log₃(50)
log(50)/log(3) = 1.699/0.477 ≈ 3.56