Log Calculator
Use this free log calculator to instantly find the logarithm of any number for any base. Whether you need a log base 10 calculator, a binary log calculator, or a logarithm calculator with base 2, 3, 5, or any custom value, just enter your inputs and get accurate results in one click.
Popular Calculators
How to Use This Logarithm Calculator?
Using this online log calculator is straightforward:
- Enter the number (exponent value) in the first field.
- Enter the log base in the second field (default is 10 for common log).
- Click Calculate to instantly get the log value.
This log calc works for all positive numbers and supports any base, making it one of the most flexible logarithm calculators available online.
What is a Logarithm? (Logarithm Explanation)
A logarithm answers a simple question: to what power must we raise a base to get a given number?
To define log in math more formally: if b^y = x, then log_b(x) = y. Here, b is the base, x is the number you want the log of, and y is the result, the exponent.
For example:
- log₁₀(1000) = 3, because 10³ = 1000
- log₂(8) = 3, because 2³ = 8
- log₅(25) = 2, because 5² = 25
This concept is fundamental to algebra, science, engineering, and computer science. Instead of pulling out a log scientific calculator or flipping through log tables, you can use this tool to compute logarithm online instantly.
Types of Logarithms
Common Logarithm (Log Base 10)
The base 10 logarithm is the most widely used in everyday math and science. Written as log(x) or log₁₀(x), it tells you how many times you multiply 10 to reach a number. Our log base 10 calculator handles all such problems instantly, no need for a log table or a log scientific calculator.
Common log is used in the Richter scale, pH calculations, decibel measurements, and more.
Natural Logarithm (Log Base e)
The natural logarithm uses base e (Euler’s number ≈ 2.71828) and is written as ln(x) or log_e(x). It appears frequently in calculus, continuous growth/decay models, and probability. This tool can serve as a natural logarithms calculator, just enter e (≈2.71828) as the base.
Binary Logarithm (Log Base 2)
The binary log calculator (log base 2) is essential in computer science and digital systems. log₂(x) tells you how many times 2 is multiplied to reach x. For instance, log₂(1024) = 10. Developers and engineers regularly need to find the log of a number in base 2 when working with binary data, algorithms, and data structures.
Logarithm Rules & Properties
Understanding log rules helps you simplify and solve complex expressions, and our log rules calculator applies all of them automatically.
Product Rule: log_b(A × B) = log_b(A) + log_b(B) Multiplication inside a log becomes addition of two separate logs.
Quotient Rule: log_b(A / B) = log_b(A) − log_b(B) Division inside a log becomes subtraction.
Power Rule: log_b(Aⁿ) = n × log_b(A) An exponent inside a log moves out as a multiplier. This is one of the most used logarithmic properties in simplification.
Change of Base Rule: log_b(A) = log_c(A) / log_c(b) This rule lets you convert to log base 10 or any convenient base when using a standard calculator. It’s the basis of the log converter feature in many tools.
Base Switch Rule: log_b(a) = 1 / log_a(b)
Log of 1: log_b(1) = 0, for any base b. Always.
Log of the Same Base: log_b(b) = 1. When base and argument are equal, the answer is always 1.
How to Calculate Log (Step-by-Step)
You can calculate log manually using these steps, or skip directly to our tool above.
Example 1: Find log₁₀(1000)
Step 1: Express 1000 as a power of 10. 1000 = 10 × 10 × 10 = 10³
Step 2: Apply the power rule. log₁₀(10³) = 3 × log₁₀(10) = 3 × 1 = 3
Example 2: Find log₂(64)
Step 1: Express 64 as a power of 2. 64 = 2⁶
Step 2: Apply the power rule. log₂(2⁶) = 6 × log₂(2) = 6 × 1 = 6
Example 3: How to calculate log base 2 when the number isn’t a clean power of 2. e.g., log₂(6)
Use the change of base rule: log₂(6) = log₁₀(6) / log₁₀(2) = 0.7782 / 0.3010 ≈ 2.585
This is exactly the kind of log solution that our log base calculator handles in a single step.
How to Find Log of a Number Using a Calculator
Many people wonder how to use log on a calculator, especially when the base isn’t 10 or e, since most physical calculators only have LOG and LN buttons.
Here’s the method to type log base 2 in a calculator (or any other base):
Use the change of base formula: log_b(x) = log(x) / log(b)
For example, to find log base 2 of 6 on a physical calculator: → Type: log(6) ÷ log(2) = 0.7782 ÷ 0.3010 ≈ 2.585
But with our online log calculator, you don’t need to remember any of this. Just enter the base and the number , and get the log value instantly.
For example, if you have the numbers 8, 3, 15, 1, and 6, their order from least to greatest would be: 1, 3, 6, 8, 15.
The opposite of least to greatest is greatest to least, which arranges the same values in descending order, from largest to smallest, or high to lowest.
Special Log Values | Quick Reference
| Expression | Value |
|---|---|
| log₁₀(1) | 0 |
| log₁₀(10) | 1 |
| log₁₀(100) | 2 |
| log₁₀(1000) | 3 |
| log₂(1) | 0 |
| log₂(2) | 1 |
| log₂(8) | 3 |
| log₂(1024) | 10 |
| ln(1) | 0 |
| ln(e) | 1 |
| log₁₀(0.05) | −1.301 |
| log₁₀(0.3) | −0.523 |
| log₁₀(1.5) | 0.176 |
| log₁₀(30) | 1.477 |
| log₁₀(300) | 2.477 |
| log₁₀(1024) | 3.010 |
Frequently Asked Questions
What is a log calculator?
A log calculator, also called a calculator of log or logarithm calculator is a tool that computes the logarithm of any number for any given base. You enter the number and the base, and it returns the log value without manual calculation or log tables.
How do I calculate log base 10?
Enter your number in the exponent field and set the base to 10. Our base 10 log calculator will return the result immediately. For example, the log base 10 calculator gives log₁₀(1000) = 3.
Can log bases be negative?
No, the base of a logarithm must always be a positive number greater than 1 (or between 0 and 1, excluding 0 and 1 exactly). A negative base leads to undefined or complex values in standard real-number math.
How do I find the log of a number without a table?
Use our log calculator online, just input the number and base. Alternatively, apply the change of base rule with any standard calculator: log_b(x) = log(x) / log(b).
What is the difference between log and ln?
“log” without a specified base typically means log base 10 (common logarithm), while “ln” means log base e (natural logarithm). Our tool handles both, and any other base.
How to do log base 5 on a calculator?
Enter 5 as the base and your number as the argument in our log base calculator. Or use the change of base formula manually: log₅(x) = log(x) / log(5).
What is log of 1?
log_b(1) = 0 for any base b. Since b⁰ = 1 always holds true, the value of log of 1 is always zero regardless of the base.
How to convert log to exponential form?
If log_b(x) = y, the exponential form is b^y = x. Our tool focuses on the log to exponential calculator direction, showing you the exponent (y) when you know the base (b) and number (x).
What is a binary log calculator used for?
The binary log calculator (log base 2) is used in computer science, information theory, algorithm analysis, and data structures. log₂(n) tells you the depth of a binary tree with n nodes, among many other applications.
How to write as a single logarithm?
Use the product, quotient, and power rules in reverse, combine multiple log terms into one expression. For example: log(2) + log(3) = log(6). Our single logarithm solver approach is built into the simplification logic.