Definition of a Logarithm
Understand what logarithms are and how they relate to exponents as inverse operations.
Definition
- Base (): The number being raised to a power (must be positive, )
- Argument (): The result of the exponentiation (must be positive)
- Logarithm (): The exponent needed
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Worked Examples
Write in logarithmic form.
Identify the base
The base is (the number being raised to a power) → Base:
Identify the exponent
The exponent is → Exponent:
Identify the result
The result of is → Argument:
Write in logarithmic form
becomes →
Answer:
Common Mistakes
Confusing the base and the argument
Why it's wrong: In , students sometimes swap and . The base is the subscript number, the argument is inside parentheses.
Correct: Remember: . The BASE is what gets raised to a power.
Thinking or 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. only exists when .
Forgetting that (no base written) means base 10
Why it's wrong: The notation can be confusing. Some textbooks use for base 10.
Correct: is the common logarithm with base 10. is the natural logarithm with base .
Writing
Why it's wrong: raised to any power always equals , so base cannot produce other values.
Correct: The base of a logarithm must be positive and not equal to : and .
Interactive Visual
Linear Function Explorer
| x | y |
|---|---|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
See how powers of a number grow on the number line. Change the base and exponent.
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y = 2x + 1
m=2, b=1
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Practice Problems
16 problemsWhat does mean in exponential form?
Why It Matters
- 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 to measure acidity.
- Computer science: Algorithm complexity often involves (binary search, sorting).
- Finance: Calculating how long to double an investment uses logarithms.
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 , and magnitude 7 has amplitude . The formula involves .
An earthquake measures magnitude 5 on the Richter scale. The Richter formula is where is amplitude.
What is the amplitude of this earthquake?
Step 1: Write the mathematical expression
If , 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 , then (acidic, like vinegar).
A solution has hydrogen ion concentration .
What is the pH of this solution?
Step 1: Write the mathematical expression
Calculate :
Key Takeaways
- 1A logarithm means (logarithms are inverse of exponents)
- 2The base () must be positive and not equal to 1
- 3The argument () must be positive
- 4Common logarithm: (base 10)
- 5Natural logarithm: (base )
- 6Key values: because , and because
Frequently Asked Questions
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 an exponential expression; appears as a subscript in logarithm notation
- Argument
- The input value of a logarithm; the number inside the parentheses in
- Common logarithm
- A logarithm with base 10, written as or
- Natural logarithm
- A logarithm with base , written as or
- Inverse operation
- An operation that reverses the effect of another; logarithms and exponentiation are inverses
Formula Card
Definition
Fundamental relationship
Log of 1
Because $b^0 = 1$
Log of base
Because $b^1 = b$
Common log
Common logarithm notation
Natural log
Natural logarithm notation