Definition of a Logarithm

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

Advanced25 minLesson
Printable worksheet

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 equals 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, b≠1b \neq 1)
  • Argument (yy): The result of the exponentiation (must be positive)
  • Logarithm (xx): The exponent needed
Example: log⁡2(8)=3\log_2(8) = 3 because 23=82^3 = 8

Try it now

What does log⁡2(8)=3\log_2(8) = 3 mean in exponential form?

Worked Examples

Write 53=1255^3 = 125 in logarithmic form.
1

Identify the base

The base is 55 (the number being raised to a power) → Base: b=5b = 5
2

Identify the exponent

The exponent is 33 → Exponent: x=3x = 3
3

Identify the result

The result of 535^3 is 125125 → Argument: y=125y = 125
4

Write in logarithmic form

log⁡b(y)=x\log_b(y) = x becomes log⁡5(125)=3\log_5(125) = 3

Common Mistakes

Confusing the base and the argument

Why it's wrong: In log⁡b(y)=x\log_b(y) = x, students sometimes swap bb and yy. The base is the subscript number, the argument is inside parentheses.

Correct: Remember: log⁡base(argument)=exponent\log_{\text{base}}(\text{argument}) = \text{exponent}. The BASE is what gets raised to a power.

Thinking log⁡b(0)\log_b(0) or log⁡b(−5)\log_b(-5) exists

Why it's wrong: No positive base raised to any power can equal zero or a negative number.

Correct: The argument of a logarithm must always be positive. log⁡b(y)\log_b(y) only exists when y>0y > 0.

Forgetting that log⁡\log (no base written) means base 10

Why it's wrong: The notation can be confusing. Some textbooks use lg⁡\lg for base 10.

Correct: log⁡(x)\log(x) is the common logarithm with base 10. ln⁡(x)\ln(x) is the natural logarithm with base e≈2.718e \approx 2.718.

Writing log⁡1(x)\log_1(x)

Why it's wrong: 11 raised to any power always equals 11, so base 11 cannot produce other values.

Correct: The base of a logarithm must be positive and not equal to 11: b>0b > 0 and b≠1b \neq 1.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Evaluate Logarithms | Step-by-Step Examples

by Understand The Math

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Interactive Visual

Function graph

051015202530-202468xy(32, 5)(8, 3)x = 0

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Asymptote

Table of values

x12481632
log₂(x)012345
2^x2416256655364294970000
10³ = 1000
3

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

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

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Try these examples:

y = log2(x)

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Practice Problems

16 problems
Problem 1 of 16
Easy
What does log⁡2(8)=3\log_2(8) = 3 mean in exponential form?

Why It Matters

Logarithms are essential tools for solving real-world problems involving exponential growth and large numbers:
  • Earthquakes: The Richter scale uses logarithms. A magnitude 7 earthquake is 10 times stronger than magnitude 6.
  • Sound: Decibels measure sound intensity logarithmically. Every 10 dB doubles perceived loudness.
  • pH levels: Chemistry uses pH=−log⁡[H+]\text{pH} = -\log[H^+] to measure acidity.
  • Computer science: Algorithm complexity often involves log⁡n\log n (binary search, sorting).
  • Finance: Calculating how long to double an investment uses logarithms.
Without logarithms, solving equations like 2x=1002^x = 100 would be nearly impossible!

Real World Applications

Measuring Earthquake Magnitude

The Richter scale uses logarithms to measure earthquake intensity. Each whole number increase represents 10 times more ground motion.

Example:

A magnitude 6 earthquake has amplitude A6A_6, and magnitude 7 has amplitude A7=10×A6A_7 = 10 \times A_6. The formula involves log⁡10\log_{10}.

1Try It Yourself

An earthquake measures magnitude 5 on the Richter scale. The Richter formula is M=log⁡10(A)M = \log_{10}(A) where AA is amplitude.

What is the amplitude AA of this earthquake?

Step 1: Write the mathematical expression

If log⁡10(A)=5\log_{10}(A) = 5, convert to exponential form:

Calculating pH in Chemistry

The pH scale measures how acidic or basic a solution is using the formula $\text{pH} = -\log[H^+]$.

Example:

If [H+]=0.001=10−3[H^+] = 0.001 = 10^{-3}, then pH=−log⁡(10−3)=−(−3)=3\text{pH} = -\log(10^{-3}) = -(-3) = 3 (acidic, like vinegar).

2Try It Yourself

A solution has hydrogen ion concentration [H+]=10−7[H^+] = 10^{-7}.

What is the pH of this solution?

Step 1: Write the mathematical expression

Calculate pH=−log⁡(10−7)\text{pH} = -\log(10^{-7}):

Key Takeaways

  • 1A logarithm log⁡b(y)=x\log_b(y) = x means bx=yb^x = y (logarithms are inverse of exponents)
  • 2The base (bb) must be positive and not equal to 1
  • 3The argument (yy) must be positive
  • 4Common logarithm: log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x) (base 10)
  • 5Natural logarithm: ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x) (base e≈2.718e \approx 2.718)
  • 6Key values: log⁡b(1)=0\log_b(1) = 0 because b0=1b^0 = 1, and log⁡b(b)=1\log_b(b) = 1 because b1=bb^1 = b

Frequently Asked Questions

If the base were negative, raising it to fractional powers would give complex numbers. If the base were 1, then 1x=11^x = 1 for all xx, so you could never get any other value. We need consistent, predictable results.
If the base were negative, raising it to fractional powers would give complex numbers. If the base were 1, then 1x=11^x = 1 for all xx, so you could never get any other value. We need consistent, predictable results.
log⁡\log (common logarithm) uses base 10 and is convenient for decimal calculations. ln⁡\ln (natural logarithm) uses base e≈2.718e \approx 2.718 and appears naturally in calculus, growth/decay problems, and continuous compounding.
No positive number raised to any real power equals zero or a negative number. For example, 2x2^x is always positive regardless of xx. So log⁡2(0)\log_2(0) and log⁡2(−5)\log_2(-5) have no real solutions.

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 an exponential expression; appears as a subscript in logarithm notation
Argument
The input value of a logarithm; the number inside the parentheses in log⁡b(y)\log_b(y)
Common logarithm
A logarithm with base 10, written as log⁡(x)\log(x) or log⁡10(x)\log_{10}(x)
Natural logarithm
A logarithm with base e≈2.718e \approx 2.718, written as ln⁡(x)\ln(x) or log⁡e(x)\log_e(x)
Inverse operation
An operation that reverses the effect of another; logarithms and exponentiation are inverses

Formula Card

Definition

log⁡b(y)=x  ⟺  bx=y\log_b(y) = x \iff b^x = y

Fundamental relationship

Log of 1

log⁡b(1)=0\log_b(1) = 0

Because $b^0 = 1$

Log of base

log⁡b(b)=1\log_b(b) = 1

Because $b^1 = b$

Common log

log⁡(x)=log⁡10(x)\log(x) = \log_{10}(x)

Common logarithm notation

Natural log

ln⁡(x)=log⁡e(x)\ln(x) = \log_e(x)

Natural logarithm notation

Today's challengeOne problem, three tries

More in This Topic

Introduction to Logarithms

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Common Logarithms (Base 10)

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Natural Logarithms (ln)

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Graphing Logarithmic Functions

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