Free Log Calculator (Logarithm) - Calculate Logarithms Online
Our free Log Calculator helps you calculate logarithms with any base. Solve for the base, argument, or result in the logarithmic equation log_b(x) = y. Supports common log (base 10), natural log (base e), binary log (base 2), and custom bases.
What is a Logarithm?
A logarithm is the inverse mathematical operation of exponentiation. In simple terms, the logarithm of a number is the exponent to which a fixed base must be raised to produce that number. The logarithmic equation is written as:
log_b(x) = y which means b^y = x
For example, log₁₀(1000) = 3 because 10³ = 1000. Logarithms are fundamental tools in mathematics, science, engineering, and computer science.
Types of Logarithms
There are three commonly used logarithm bases:
- Common Logarithm (base 10) — Written as log(x) or log₁₀(x). Used in science, engineering, and the Richter scale for earthquake magnitudes.
- Natural Logarithm (base e) — Written as ln(x) or logₑ(x). The base e (Euler's number, approximately 2.71828) is used extensively in calculus, physics, and financial mathematics.
- Binary Logarithm (base 2) — Written as log₂(x). Used in computer science for analyzing algorithm complexity, information theory, and digital systems.
How to Use the Log Calculator
Calculate the Result (y)
- Select a base (10, e, 2, or enter a custom value)
- Enter the argument (the number x)
- The calculator instantly computes log_b(x)
Calculate the Base (b)
- Choose "Solve for Base"
- Enter the argument (x) and the result (y)
- The calculator finds the base that satisfies the equation
Calculate the Argument (x)
- Choose "Solve for Argument"
- Enter the base (b) and the result (y)
- The calculator computes b^y
Features of Our Log Calculator
- Any base supported — Use base 10, e, 2, or enter any custom base
- Three solve modes — Calculate the result, base, or argument
- Step-by-step solutions — See exactly how the answer was computed
- Verification — Built-in check confirms your result is correct
- Common log table — Quick reference for frequently used logarithm values
- Log rules reference — Built-in guide to key logarithm properties
- Real-time calculation — Results update instantly as you type
Logarithm Rules and Properties
Understanding these fundamental rules will help you work with logarithms more effectively:
Product Rule
log_b(x × y) = log_b(x) + log_b(y)
The logarithm of a product equals the sum of the logarithms of the factors.
Example: log(1 × 10) = log(1) + log(10) = 0 + 1 = 1
Quotient Rule
log_b(x / y) = log_b(x) − log_b(y)
The logarithm of a quotient equals the difference of the logarithms.
Example: log(10 / 2) = log(10) − log(2) = 1 − 0.301 = 0.699
Power Rule
log_b(x^y) = y × log_b(x)
The logarithm of a number raised to a power equals the exponent times the logarithm of the number.
Example: log(2⁶) = 6 × log(2) = 1.806
Change of Base Formula
log_b(x) = log_k(x) / log_k(b)
This allows you to convert a logarithm from one base to another. Our calculator uses this formula internally with the natural log (ln) for computation.
Special Values
| Expression | Value |
|---|---|
| log_b(1) | 0 (any base) |
| log_b(b) | 1 (any base) |
| log₁₀(10) | 1 |
| ln(e) | 1 |
| log₁₀(100) | 2 |
| log₂(8) | 3 |
| ln(1) | 0 |
Real-World Applications of Logarithms
Science and Engineering
Logarithms are used to express quantities that span many orders of magnitude. The Richter scale (earthquake intensity), pH scale (acidity), and decibel scale (sound intensity) are all logarithmic scales.
Computer Science
Binary logarithms (base 2) are essential for analyzing algorithm complexity. For example, binary search runs in O(log₂n) time, making it extremely efficient for large datasets.
Finance
Natural logarithms appear in continuous compound interest calculations and the Black-Scholes model for options pricing.
Music
Musical pitch is logarithmic — each octave represents a doubling of frequency. The relationship between notes can be expressed using logarithms.
FAQs About Log Calculator
What is the difference between log and ln?
"log" typically refers to the common logarithm with base 10, while "ln" refers to the natural logarithm with base e (Euler's number, approximately 2.71828). Both are logarithms with different bases.
Can I use a negative number in a logarithm?
No, logarithms are only defined for positive real numbers. The domain of log_b(x) requires x > 0 and b > 0 with b ≠ 1.
What is the change of base formula?
The change of base formula allows you to compute logarithms in any base using a different base: log_b(x) = ln(x) / ln(b). This is how most calculators compute logarithms internally.
Why is log(0) undefined?
There is no number y such that b^y = 0 for any positive base b. As x approaches 0, log(x) approaches negative infinity, but it never actually reaches 0.
How do I convert between log bases?
Use the change of base formula: log_b(x) = log_k(x) / log_k(b), where k is any positive base (typically e or 10).
What is Euler's number (e)?
Euler's number e is an irrational constant approximately equal to 2.71828. It is the base of the natural logarithm and arises naturally in calculus, particularly in problems involving growth and decay.
Why Use Our Log Calculator?
- Free and instant — No registration, no downloads, no waiting
- Accurate results — High-precision calculations up to 6 decimal places
- Educational — Step-by-step solutions help you learn
- Privacy focused — All calculations run in your browser
- Mobile friendly — Works perfectly on phones, tablets, and desktops
- Works offline — Use the calculator even without an internet connection
Start calculating logarithms now with our free online Log Calculator!