Introduction to Logarithms

Learn what logarithms are and how they relate to exponents as inverse operations.

Advanced25 minLesson
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Definition

A logarithm answers the question: "What exponent do I need?"
If , then
In words: **"The logarithm base of is " means " raised to the power equals "**
Key components:
  • Base (): The number being raised to a power (must be positive, not equal to 1)
  • Argument (): The result we're taking the log of (must be positive)
  • Exponent (): The answer to the logarithm
Special logarithms:
  • Common logarithm: means (base 10)
  • Natural logarithm: means (base )

Try it now

What does equal?

Worked Examples

Write in logarithmic form.

1

Identify the base

The base is (the number being raised to a power) → Base =

2

Identify the exponent

The exponent is → Exponent =

3

Identify the result

The result is → Result =

4

Write in log form:

→

Common Mistakes

Confusing the base and the argument

Why it's wrong: In , students often mix up which number is the base and which is the argument.

Correct: The base is the small subscript number (). The argument is inside the parentheses (). Memory tip: The base goes at the bottom (subscript).

Thinking

Why it's wrong: This is FALSE! Logarithms do not distribute over addition.

Correct: The correct rule is . Only multiplication inside the log becomes addition outside.

Forgetting that without a base means base 10

Why it's wrong: Students sometimes assume no base means base or base 2.

Correct: (common log). For base , we write .

Trying to take the logarithm of a negative number or zero

Why it's wrong: The argument of a logarithm must be positive. and are undefined.

Correct: Always check that the argument is positive before evaluating. requires .

Interactive Visual

2^3 = 8
3

See how powers of a number grow on the number line. Change the base and exponent.

Linear Function Explorer

y = x
Slope (m)1
Y-Intercept (b)0
b
run
rise
xy
-2-2
-1-1
00
11
22

Interactive Sandbox

Interactive Grapher

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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History

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Practice Problems

17 problems
Problem 1 of 17
Easy

What does equal?

Why It Matters

Logarithms are essential tools in mathematics, science, and everyday life:
  • Earthquake measurement: The Richter scale uses logarithms. An earthquake of magnitude 6 is 10 times stronger than magnitude 5!
  • Sound intensity: Decibels measure sound on a logarithmic scale
  • pH levels: The acidity of a solution is measured using
  • Population growth: Scientists use logarithms to model exponential growth and decay
  • Computer science: Algorithm complexity often involves
  • Finance: Compound interest and investment growth use logarithmic calculations
Without logarithms, we couldn't easily work with very large numbers or solve exponential equations!

Real World Applications

Earthquake Magnitude (Richter Scale)

The Richter scale measures earthquake intensity using logarithms. Each whole number increase represents a 10-fold increase in amplitude.

Example:

A magnitude 7 earthquake is times more powerful than a magnitude 5 earthquake.

1Try It Yourself

An earthquake measures 6.0 on the Richter scale. Another measures 4.0.

How many times more powerful is the first earthquake?

Step 1: Write the mathematical expression

Calculate :

Sound Intensity (Decibels)

Sound intensity is measured in decibels (dB), which uses a logarithmic scale. The formula is $dB = 10 \log\left(\frac{I}{I_0}\right)$.

Example:

A sound that is 1000 times more intense than the reference level has intensity dB.

2Try It Yourself

A lawn mower produces sound at 90 dB. A whisper is 30 dB.

How many times more intense is the lawn mower sound?

Step 1: Write the mathematical expression

Calculate :

Algorithm Complexity in Computer Science

Many efficient algorithms have logarithmic time complexity, written as $O(\log n)$. This is why binary search is so fast!

Example:

To find a word in a dictionary of 1024 pages using binary search, you need at most steps.

3Try It Yourself

A sorted database has 1,000,000 records. You use binary search.

Approximately how many comparisons are needed to find any record?

Step 1: Write the mathematical expression

Calculate :

Key Takeaways

  • 1A logarithm answers: "What exponent gives this result?" If , then
  • 2 means (common log); means (natural log)
  • 3The base must be positive and not equal to 1; the argument must be positive
  • 4 for any valid base (because )
  • 5 for any valid base (because )
  • 6Logarithms and exponents are inverse operations

Frequently Asked Questions

Because no real exponent makes a positive base equal zero or negative. For example, there's no where or .
Because no real exponent makes a positive base equal zero or negative. For example, there's no where or .
typically means (base 10), used in science and engineering. means (base ), used in calculus and natural growth/decay.
Because any number raised to the power 0 equals 1. So means for any valid base .

Glossary

Logarithm
The exponent to which a base must be raised to produce a given number: means
Base
The number being raised to a power in a logarithm; written as subscript in
Argument
The number inside the logarithm; the value we're taking the log of
Common logarithm
Logarithm with base 10, written as or
Natural logarithm
Logarithm with base , written as or
Inverse operations
Operations that undo each other; logarithms and exponentiation are inverses

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Definition of a Logarithm

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