Log Calculator
Use this log calculator to evaluate common, natural, binary, and custom-base logarithms, view equivalent exponential form, and follow clear calculation steps online.
Calculate logb(x) = y
To calculate the logarithm of a number x with base b, enter any two values below and leave the third blank.
Use 10 (common), e (natural), 2 (binary), or any base > 0 and ≠ 1.
Enter values above to see results
About This Tool
This log calculator finds the exponent to which a selected base must be raised to produce a positive number. Choose base 10, base e, base 2, or enter another valid base, then review the result in logarithmic and exponential form.
The tool also works as a logarithm calculator for decimal values and large or small positive numbers. It is intended for evaluating logarithms rather than solving complete logarithmic equations with variables.
How to Use the Log Calculator
- Enter a positive number in the argument field.
- Select common log, natural log, binary log, or a custom base. For a custom base, enter a value greater than 0 and not equal to 1.
- Click calculate.
- Review the result and equivalent exponential form.
- Check the displayed steps or change-of-base working.
Before starting a log calculation, confirm that the argument and base satisfy the real-number input rules.
What Is a Logarithm?
A logarithm tells you which exponent must be applied to a base to obtain a given number. In symbols, logb(x) = y means by = x.
For example, log₁₀(1000) = 3 because 10³ = 1000. The calculator uses this inverse relationship between logarithms and exponents to find the missing exponent.
Types of Logarithms
Common Logarithm
A common logarithm uses base 10 and is often written as log(x) without showing the base. Example: log₁₀(100) = 2.
Natural Logarithm
A natural logarithm uses the mathematical constant e as its base and is written as ln(x). Example: ln(e4) = 4.
Binary Logarithm
A binary logarithm uses base 2 and is frequently used in computer science and information theory. Example: log₂(32) = 5.
A logarithm calculator with preset bases makes it easier to switch between these common forms without manually entering each base.
How to Calculate Log Values
When the result is not an obvious whole number, use the change-of-base formula:
logb(x) = ln(x) ÷ ln(b)
You may also use common logarithms: logb(x) = log(x) ÷ log(b).
For example, to calculate log base 5 of 125: log₅(125) = ln(125) ÷ ln(5) = 3. The tool applies this method automatically for valid custom bases.
Log Calculation Examples
| Expression | Result | Equivalent Exponential Form |
|---|---|---|
| log₁₀(1000) | 3 | 10³ = 1000 |
| log₂(32) | 5 | 2⁵ = 32 |
| ln(e4) | 4 | e4 = e4 |
| log₅(125) | 3 | 5³ = 125 |
| log₁₀(0.01) | -2 | 10⁻² = 0.01 |
These examples show that logarithm results can be positive, zero, negative, or decimal values.
Basic Logarithm Rules
| Rule | Formula |
|---|---|
| Product Rule | logb(xy) = logb(x) + logb(y) |
| Quotient Rule | logb(x ÷ y) = logb(x) − logb(y) |
| Power Rule | logb(xn) = n · logb(x) |
These rules are useful for rewriting or simplifying logarithmic expressions. They do not change the base restrictions or allow a real logarithm of zero or a negative argument.
Valid Inputs and Restrictions
For real-number logarithms:
- The argument must be greater than 0.
- The base must be greater than 0.
- The base cannot equal 1.
A base of 1 is invalid because every power of 1 equals 1, so it cannot produce the full range of positive values. A negative base is not supported by a standard real-number logarithm tool.
Why Is the Logarithm of Zero Undefined?
No valid positive base raised to a finite real exponent equals zero. Therefore, the logarithm of zero has no finite real value. A real logarithm of a negative number is also undefined. Complex logarithms follow different rules and are outside the scope of this tool.
Logarithm vs Exponentiation
Exponentiation starts with a base and exponent to find a value: 2⁵ = 32. A logarithm reverses that process: log₂(32) = 5. This inverse process moves from the known base and result back to the exponent.
Logarithm vs Antilogarithm
A logarithm finds an exponent. An antilogarithm uses the exponent and base to recover the original number. For example, log₁₀(100) = 2, and the corresponding antilogarithm is 10² = 100. Use an antilogarithm tool when the exponent is known and you need the original value.
Log Evaluation vs Logarithmic Equation Solving
Evaluating a logarithm and solving a logarithmic equation are different tasks. Evaluate: log₂(32) gives 5. Solve: log₂(x + 3) = 4 gives x = 13.
This log calculator focuses on evaluating numerical inputs. Use a logarithmic equation calculator when the expression contains an unknown variable that must be isolated.
Common Input Mistakes
Avoid these errors:
- Entering zero or a negative argument
- Selecting a base of 1
- Entering a negative base
- Confusing log with ln
- Reversing the argument and the base
- Entering an equation into a basic evaluator
- Rounding intermediate values too early
- Applying log rules to addition or subtraction incorrectly
Related Calculators
You may also find these tools useful:
- Scientific Notation Calculator
- Simplify Calculator
- Natural Logarithm Toolcoming soon
- Antilogarithm Toolcoming soon
- Logarithmic Equation Calculatorcoming soon
- Exponent Calculatorcoming soon
- Exponential Growth Calculatorcoming soon
- Scientific Calculatorcoming soon
Start Calculating
Enter a positive number, choose a valid base, and use the log calculator to review the logarithm value, equivalent exponential form, and calculation steps.
Frequently Asked Questions
What does a log calculator do?
It finds the exponent required to raise a selected base to an entered positive number.
What is the difference between log and ln?
Log commonly refers to base 10, while ln refers to the natural logarithm with base e.
Can the base be any number?
For a real logarithm, the base must be positive and cannot equal 1.
Why is my answer a decimal?
Many numbers are not exact powers of the selected base, so their logarithms are non-integer decimal values.
Can this tool solve logarithmic equations?
It is primarily an evaluator. Equations containing unknown variables should be entered into a dedicated equation-solving tool.