Common Logarithm
A common logarithm uses base 10. For example, log₁₀(1000) equals 3 because 10³ equals 1000.
Advanced Math Tool
Logarithm Calculator
Calculate common logarithms, natural logarithms, binary logarithms, and logarithms with any valid custom base.
Calculator
Calculate a Logarithm
Enter a positive number and select the logarithm base you want to use.
Result
Your Answer
The calculated logarithm and conversion formula will appear below.
Enter a number, choose a base, and click “Calculate Logarithm” to view your answer.
About the calculator
What Is a Logarithm Calculator?
A logarithm calculator finds the exponent to which a base must be raised to produce a particular number.
Natural Logarithm
A natural logarithm uses Euler’s number, approximately 2.71828, as its base and is normally written as ln(x).
Binary Logarithm
A binary logarithm uses base 2 and is commonly used in computer science, algorithms, and information theory.
Logarithm Formula
The statement log base b of x equals y means that b raised to the power y produces x.
logb(x) = y
Equivalent exponential form: by = x
Simple instructions
How to Use the Logarithm Calculator
Calculate any valid logarithm in three straightforward steps.
1
Enter the Number
Type the positive number for which you want to calculate the logarithm.
2
Select a Base
Choose base 10, base e, base 2, or enter your own custom logarithm base.
3
View the Result
Click the calculate button to see the logarithm value and the formula used.
Practical reference
Common Logarithm Examples
These examples demonstrate how logarithms relate to exponents.
| Expression | Result | Explanation |
|---|---|---|
log₁₀(100) |
2 | 10² = 100 |
log₂(8) |
3 | 2³ = 8 |
log₅(125) |
3 | 5³ = 125 |
ln(e) |
1 | e¹ = e |
log₁₀(0.1) |
-1 | 10⁻¹ = 0.1 |
Frequently asked questions
Logarithm Calculator FAQs
Find clear answers to common questions about logarithms and their valid values.
A logarithm calculates the exponent needed to raise a selected base to obtain a given number. For example, log₂(8) is 3 because 2³ equals 8.
Log commonly refers to a base-10 logarithm, while ln refers to the natural logarithm with base e, approximately equal to 2.71828.
Yes. When a number is between 0 and 1 and the base is greater than 1, its logarithm is negative. For example, log₁₀(0.01) equals -2.
One raised to any exponent always remains one. Therefore, base 1 cannot produce other positive numbers and does not define a useful logarithm.
No. A real-number logarithm is only defined for positive input values. Zero and negative numbers do not have real logarithm values.
The calculator uses the change-of-base formula: log base b of x equals ln(x) divided by ln(b). This works for every valid positive base except 1.