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Mom's Basement University

Introduction to Logarithms

What You’ll Learn

Section titled “What You’ll Learn”

In this lesson you’ll learn the definition of a logarithm, how logarithms are the inverse of exponentials, and how to convert between logarithmic and exponential forms.

The Concept

Section titled “The Concept”

A logarithm asks the question: “To what power must the base be raised to get this number?”

The logarithmic form log_b(a) = c means:

bc=ab^c = a

The graph shows the relationship visually. The green curve is the exponential y = 2^x, which grows rapidly. The blue curve is its inverse, y = log₂(x), which grows slowly. They are mirror images across the dashed y = x line. The gold dot at (1, 0) shows that log₂(1) = 0, because 2⁰ = 1.

Key facts:

  • The base b must be positive and not equal to 1.
  • The argument a must be positive (you can’t take the log of zero or a negative).
  • Logarithms and exponentials are inverses: log_b(b^x) = x and b^(log_b(x)) = x.

Common bases:

  • Base 10 (common logarithm): written as “log” without a subscript. This is the default on most calculators. It’s used for anything involving powers of 10: the Richter scale, decibels, pH, and scientific notation all use base 10. When you see “log(100)” with no base written, it means log₁₀(100) = 2.

  • Base e ≈ 2.718 (natural logarithm): written as “ln”. The number e is a special irrational constant that shows up naturally in continuous growth and decay: things like population growth, radioactive decay, compound interest compounded continuously, and calculus. When you see “ln(x)”, it means log_e(x). You’ll encounter e more in Pre-Calculus and Calculus, but it’s good to know it exists now.

  • Base 2 (binary logarithm): written as log₂. This is the one we graphed above. It’s especially important in computer science, where everything is built on powers of 2 (bits, bytes, binary).

Any positive number (except 1) can be a base, but these three come up the most. In Algebra 2, you’ll mostly work with base 10 and general bases. The natural logarithm becomes central in Pre-Calculus and beyond.

Worked Example

Section titled “Worked Example”

1. Convert to exponential form: log₂(8) = 3

23=82^3 = 8

2. Convert to logarithmic form: 10⁴ = 10000

log⁡10(10000)=4\log_{10}(10000) = 4

3. Evaluate log₅(125)

Ask: 5 to what power equals 125? Since 5³ = 125:

log⁡5(125)=3\log_5(125) = 3

4. Solve for x: log₃(x) = 4

Convert to exponential form:

34=x⇒x=813^4 = x \quad\Rightarrow\quad x = 81

Real-World Application

Section titled “Real-World Application”

Logarithms are used in many practical areas:

  • Earthquakes: the Richter scale is logarithmic (each whole number is 10 times stronger).
  • Sound: decibels measure sound intensity on a logarithmic scale.
  • pH scale: acidity is measured logarithmically.
  • Finance: compound interest problems often use logarithms to solve for time.
  • Computer science: logarithms measure algorithm efficiency (e.g., O(log n)).

Example: if an investment grows exponentially, logarithms help you solve for how many years it takes to reach a target amount.

Quiz

Section titled “Quiz”
The equation $\log_3(27) = 3$ means:
Convert $5^2 = 25$ to logarithmic form:
What is $\log_{10}(1000)$?
Logarithms and exponentials are:
Evaluate $\log_2(32)$.