Use this free Logarithm Calculator to find the logarithm of any positive number quickly and easily. Choose base 10, base 2, the natural logarithm (base e), or enter a custom base. Get instant results with adjustable decimal precision, making it useful for students, teachers, and anyone solving math problems.
Logarithm Calculator
Calculate logarithms of positive numbers using base 10, base e, base 2, or any custom base. Results update automatically.
Try an example
Select an example to load its values and calculate the logarithm.
Calculation Result
View your logarithm result, formula, and calculation details here.
Logarithm Result
2.00000000
log₁₀(100) = 2.00000000
Calculation complete.
Calculation details
Common logarithm rules
logb(1) = 0 for every valid base b.
logb(b) = 1 for every valid base b.
logb(x × y) = logb(x) + logb(y).
logb(x / y) = logb(x) − logb(y).
A valid real logarithm requires x > 0, b > 0, and b ≠ 1.
What Is a Logarithm?
A logarithm tells you the power to which a number, called the base, must be raised to get another number. In simple terms, logarithms are the reverse of exponentiation.
For example, log₁₀(100) = 2 because 10² = 100. Similarly, log₂(8) = 3 because 2³ = 8.
Logarithms are commonly used in mathematics, science, engineering, finance, and computer science to solve equations and understand quantities that change exponentially.
Logarithm Formula
The general logarithm formula is:
logᵦ(x) = y
This means:
bʸ = x
Here, b is the base, x is the positive number, and y is the logarithm.
For real-valued logarithms, the number must be greater than zero, and the base must be positive and cannot equal 1.
Types of Logarithms
There are several common types of logarithms:
- Common logarithm: Uses base 10 and is written as log₁₀(x). For example, log₁₀(1000) = 3.
- Natural logarithm: Uses Euler’s number e as its base, approximately 2.71828. It is written as ln(x).
- Binary logarithm: Uses base 2 and is frequently used in computer science. For example, log₂(16) = 4.
- Custom-base logarithm: Uses any valid base greater than zero except 1.
How to Use the Logarithm Calculator
Follow these simple steps to calculate a logarithm:
- Enter the positive number you want to calculate.
- Select the logarithm type from the dropdown menu.
- If you choose Custom Base, enter your preferred base.
- Select the desired decimal precision.
- View your result and calculation details instantly.
You can also select one of the example buttons to try a calculation. Use Clear All to start again or Reload Defaults to restore the original settings.
Common Logarithm Examples
Here are some useful logarithm values:
| Calculation | Answer |
|---|---|
| log₁₀(1) | 0 |
| log₁₀(10) | 1 |
| log₁₀(100) | 2 |
| log₁₀(1000) | 3 |
| log₂(8) | 3 |
| log₂(16) | 4 |
| ln(e) | 1 |
| ln(1) | 0 |
These examples show how logarithms help identify the exponent needed to produce a particular number.
Where Are Logarithms Used?
Logarithms have many practical applications. They are used to solve exponential equations, analyze population growth, calculate compound interest, study scientific data, and measure quantities across different scales.
In computer science, binary logarithms help explain algorithms and data structures. In science and engineering, logarithmic scales are useful for representing values that vary over a wide range.
Whether you are studying mathematics or checking a calculation, a logarithm calculator can save time and help you verify your answers.
Frequently Asked Questions
1. What is a logarithm?
A logarithm tells you the exponent to which a base must be raised to obtain a given positive number. For example, log₁₀(100) = 2 because 10² = 100.
2. How do I calculate a logarithm?
Enter a positive number, select the logarithm base, and the calculator will display the result automatically. You can also choose the number of decimal places for your answer.
3. What is the difference between log and ln?
The notation log often represents a logarithm with base 10 in school mathematics, while ln specifically represents a natural logarithm with base e, approximately 2.71828.
4. Can I calculate logarithms with any base?
Yes. This calculator supports base 10, base 2, base e, and custom bases. For real-valued logarithms, the base must be positive and cannot equal 1.
5. Why can't I calculate the logarithm of zero or a negative number?
Real-valued logarithms are defined only for positive numbers. Therefore, zero and negative inputs do not produce real logarithm results.