📌 What is a Logarithm?
A logarithm answers the question: "To what power must I raise a base number to get another number?"
Logarithmic form: logb(x) = y means by = x
Example: log2(8) = 3 because 2³ = 8
🔁 Relationship with Exponents
Logarithms are the inverse of exponents. This means they "undo" each other:
log10(1000) = 3← exponent form:10³ = 1000log5(25) = 2← exponent form:5² = 25
📚 Common Types of Logarithms
- Common Logarithm (Base 10):
log(x)— Default in calculators - Natural Logarithm (Base e ≈ 2.718):
ln(x)— Used in calculus, growth models - Binary Logarithm (Base 2):
log2(x)— Used in computer science
📐 Properties and Rules of Logarithms
These rules make it easier to work with logarithmic expressions:
- 1. Product Rule:
logb(xy) = logb(x) + logb(y) - 2. Quotient Rule:
logb(x/y) = logb(x) - logb(y) - 3. Power Rule:
logb(xⁿ) = n × logb(x) - 4. Change of Base Formula:
logb(x) = logk(x) / logk(b)(commonly rewritten using base 10 or base e)
Examples:
log2(32) = log2(2⁵) = 5log3(81) = log3(3⁴) = 4
📊 Graphing Logarithmic Functions
The basic logarithmic graph has the form y = logb(x).
- It passes through (1, 0) — because
logb(1) = 0 - Increases slowly for large x
- Never touches the y-axis (asymptote at x = 0)
As the base increases, the graph becomes flatter.
🌍 Applications of Logarithms
- Data Science: Log transforms for skewed data
- Computer Science: Algorithmic complexity like
O(log n) - Finance: Continuous compounding:
A = Pert - Science: pH level (
-log[H⁺]), sound (decibels), Richter scale (earthquakes) - Information Theory: Entropy uses log base 2
🧠 Practice Questions
- Evaluate:
log2(16) - Simplify:
log10(100) + log10(1000) - Use the power rule:
log3(x⁴) - Convert to exponential form:
log4(64) = 3