Introduction to Logarithms

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

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Definition

A logarithm answers the question: "What exponent do I need?"
If bx=yb^x = y, then log⁡b(y)=x\log_b(y) = x
In words: **"The logarithm base bb of yy is xx" means "bb raised to the power xx equals yy"**
log⁡b(y)=x  ⟺  bx=y\log_b(y) = x \iff b^x = y
Key components:
  • Base (bb): The number being raised to a power (must be positive, not equal to 1)
  • Argument (yy): The result we're taking the log of (must be positive)
  • Exponent (xx): The answer to the logarithm
Special logarithms:
  • Common logarithm: log⁡(x)\log(x) means log⁡10(x)\log_{10}(x) (base 10)
  • Natural logarithm: ln⁡(x)\ln(x) means log⁡e(x)\log_e(x) (base e≈2.718e \approx 2.718)

Try it now

What does log⁡2(8)\log_2(8) equal?

Worked Examples

Write 25=322^5 = 32 in logarithmic form.

1

Identify the base

The base is 22 (the number being raised to a power) → Base = 22

2

Identify the exponent

The exponent is 55 → Exponent = 55

3

Identify the result

The result is 3232 → Result = 3232

4

Write in log form: log⁡base(result)=exponent\log_{\text{base}}(\text{result}) = \text{exponent}

log⁡2(32)=5\log_2(32) = 5

Common Mistakes

Confusing the base and the argument

Why it's wrong: In log⁡b(x)\log_b(x), students often mix up which number is the base and which is the argument.

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

Thinking log⁡(x+y)=log⁡(x)+log⁡(y)\log(x + y) = \log(x) + \log(y)

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

Correct: The correct rule is log⁡(x⋅y)=log⁡(x)+log⁡(y)\log(x \cdot y) = \log(x) + \log(y). Only multiplication inside the log becomes addition outside.

Forgetting that log⁡\log without a base means base 10

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

Correct: log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x) (common log). For base ee, we write ln⁡(x)\ln(x).

Trying to take the logarithm of a negative number or zero

Why it's wrong: The argument of a logarithm must be positive. log⁡(−5)\log(-5) and log⁡(0)\log(0) are undefined.

Correct: Always check that the argument is positive before evaluating. log⁡b(x)\log_b(x) requires x>0x > 0.

Interactive Visual

2^3 = 8
3

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Practice Problems

17 problems
Problem 1 of 17
Easy

What does log⁡2(8)\log_2(8) 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 pH=−log⁡[H+]\text{pH} = -\log[H^+]
  • Population growth: Scientists use logarithms to model exponential growth and decay
  • Computer science: Algorithm complexity often involves log⁡2(n)\log_2(n)
  • 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 107−5=10010^{7-5} = 100 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 106−410^{6-4}:

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 10log⁡(1000)=10×3=3010 \log(1000) = 10 \times 3 = 30 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 10(90−30)/1010^{(90-30)/10}:

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 log⁡2(1024)=10\log_2(1024) = 10 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 log⁡2(1000000)\log_2(1000000):

Key Takeaways

  • 1A logarithm answers: "What exponent gives this result?" If bx=yb^x = y, then log⁡b(y)=x\log_b(y) = x
  • 2log⁡(x)\log(x) means log⁡10(x)\log_{10}(x) (common log); ln⁡(x)\ln(x) means log⁡e(x)\log_e(x) (natural log)
  • 3The base must be positive and not equal to 1; the argument must be positive
  • 4log⁡b(1)=0\log_b(1) = 0 for any valid base bb (because b0=1b^0 = 1)
  • 5log⁡b(b)=1\log_b(b) = 1 for any valid base bb (because b1=bb^1 = b)
  • 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 xx where 10x=010^x = 0 or 10x=−510^x = -5.
Because no real exponent makes a positive base equal zero or negative. For example, there's no xx where 10x=010^x = 0 or 10x=−510^x = -5.
log⁡\log typically means log⁡10\log_{10} (base 10), used in science and engineering. ln⁡\ln means log⁡e\log_e (base e≈2.718e \approx 2.718), used in calculus and natural growth/decay.
Because any number raised to the power 0 equals 1. So b0=1b^0 = 1 means log⁡b(1)=0\log_b(1) = 0 for any valid base bb.

Glossary

Logarithm
The exponent to which a base must be raised to produce a given number: log⁡b(y)=x\log_b(y) = x means bx=yb^x = y
Base
The number being raised to a power in a logarithm; written as subscript in log⁡b\log_b
Argument
The number inside the logarithm; the value we're taking the log of
Common logarithm
Logarithm with base 10, written as log⁡(x)\log(x) or log⁡10(x)\log_{10}(x)
Natural logarithm
Logarithm with base e≈2.718e \approx 2.718, written as ln⁡(x)\ln(x) or log⁡e(x)\log_e(x)
Inverse operations
Operations that undo each other; logarithms and exponentiation are inverses

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