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The Complete Guide to Logarithms: Rules, Examples, and Applications
Published 15 March 2026 · 9 min read
Logarithms are one of the most important mathematical tools in science, engineering, and finance. They reverse the operation of exponentiation, turning multiplication into addition and making it possible to solve equations where the unknown appears as an exponent. From measuring earthquake intensity to calculating pH levels, logarithms are everywhere.
This guide covers everything you need to know about logarithms: the definition, the core rules, how to change bases, and real-world applications. Each section includes worked examples so you can build confidence with this essential tool.
What Is a Logarithm?
A logarithm answers the question: “To what power must we raise the base to get a given number?” The logarithm is the inverse of exponentiation.
Definition
In words: the log base of equals , which means raised to the power gives .
Common Bases
Common log (base 10)
Used in pH calculations, decibels, and the Richter scale.
Natural log (base e)
Used in calculus, continuous growth models, and probability.
Binary log (base 2)
Used in computer science, information theory, and algorithm analysis.
The Three Core Logarithm Rules
These rules follow directly from the laws of exponents and are essential for simplifying logarithmic expressions.
1. Product Rule
The log of a product equals the sum of the logs. This is why logarithms were historically used to simplify multiplication.
Example:
2. Quotient Rule
The log of a quotient equals the difference of the logs.
Example:
3. Power Rule
The log of a power equals the exponent times the log of the base.
Example:
The Change of Base Formula
Most calculators only have buttons for (base 10) and (base ). To compute a logarithm with any other base, use the change of base formula:
where can be any base (typically 10 or ).
Example: Compute
This makes sense because .
Worked Examples
Example 1: Simplifying a Logarithmic Expression
Simplify .
Using the product and quotient rules:
Example 2: Solving an Exponential Equation
Solve .
Take the logarithm of both sides:
Apply the power rule:
Example 3: Expanding a Logarithmic Expression
Expand .
Apply the quotient rule, then the product rule:
Apply the power rule (noting that ):
Real-World Applications
The Richter Scale (Earthquakes)
The Richter scale measures earthquake magnitude on a logarithmic scale:
Each whole number increase represents a tenfold increase in measured amplitude. A magnitude 6 earthquake is 10 times stronger than magnitude 5, and 100 times stronger than magnitude 4.
pH Scale (Chemistry)
The pH of a solution is defined as the negative logarithm of the hydrogen ion concentration:
A solution with has a pH of 3 (acidic), while gives pH 7 (neutral).
Decibels (Sound)
Sound intensity is measured in decibels using a logarithmic formula:
A 10 dB increase represents a tenfold increase in sound intensity. Normal conversation is about 60 dB, while a rock concert can reach 110 dB.
Important Identities
- because
- because
- (the log and exponent cancel)
- (the exponent and log cancel)
- and
Common Mistakes
Treating log(a + b) as log(a) + log(b)
The product rule says , but there is no simple rule for the log of a sum. .
Dividing logs incorrectly
. The quotient rule applies to the log of a fraction, not to a fraction of logs.
Forgetting the domain
Logarithms are only defined for positive arguments. and are undefined in real numbers.
Confusing ln and log
In many textbooks, means base 10, while means base . In some contexts (especially higher mathematics), means the natural logarithm. Always check the convention being used.
Try It Yourself
Ready to compute logarithms? Our free log calculator handles common logs, natural logs, and logs with any base. Enter your values and get instant results.
Open Log CalculatorRelated Articles
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