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Logarithm Calculator.
Calculate a logarithm for any valid positive base and argument.
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Results update as you type.
Greater than 0 and not 1
How to Use
A logarithm answers the question: what exponent on the base gives the argument? For example, log₂(8) = 3 because 2³ = 8.
For a real logarithm, the argument must be greater than 0, and the base must be greater than 0 but not equal to 1. Bases between 0 and 1 are valid and produce a decreasing logarithm.
FAQs
What is a logarithm?
A logarithm is the exponent needed to raise a base to a given positive argument. For example, log₂(8) = 3 because 2³ = 8.
Which bases are valid?
For real logarithms, the base must be positive and different from 1, and the argument must be greater than 0.
How to Use the Logarithm Calculator
Enter a positive argument and a valid base to calculate an arbitrary-base logarithm.
- Enter the positive argument x.
- Enter a base greater than 0 and different from 1.
- Review the change-of-base steps and result.
Change-of-Base Formula
The calculator evaluates arbitrary bases using natural logarithms.
Logarithm
The real logarithm requires x > 0, b > 0, and b ≠ 1.
Example:
Input: x = 8, b = 2
Calculation: ln(8) ÷ ln(2)
Result: 3
Common Use Cases
Logarithms appear in mathematics, science, engineering, and models of exponential change.
Algebra and calculus
Check logarithm identities and solve exponential relationships.
Scientific scales
Work with quantities expressed on logarithmic scales.
Domain Rules
Tips
- The argument must be greater than zero.
- The base must be positive and cannot equal one.
- Bases between zero and one produce a decreasing logarithm.
Common Mistakes to Avoid
- Entering zero or a negative argument for a real logarithm.
- Confusing the base with the argument.
Related Calculators
Cite this calculator
Canonical URL: https://mathify.one/en/math/logarithm
Cite as: Mathify. (2026). Logarithm Calculator. https://mathify.one/en/math/logarithm
Use Cases
Solve math homework problems
Quickly find logarithms with any base for algebra or precalculus assignments, with step-by-step working to verify your own calculations.
Example: Find log base 2 of 8: enter base 2 and number 8 to get 3.
Check scientific or engineering calculations
Use the calculator to verify logarithms in fields like computer science (log base 2), acoustics (decibels), or chemistry (pH), ensuring accuracy.
Example: Compute log base 10 of 1000 to confirm a pH-related value.
Frequently Asked Questions
- How does the Logarithm Calculator work?
- It uses the change-of-base formula: log_b(x) = log(x) / log(b). You input the base (b) and the number (x), and the calculator computes the logarithm, showing each step of the calculation.
- What is the change-of-base formula?
- The change-of-base formula allows you to compute logarithms with any base using common (base 10) or natural (base e) logarithms: log_b(x) = log_c(x) / log_c(b). This calculator uses it to provide results for any positive base and number.
- Can I use this calculator for any base?
- Yes, as long as the base is a positive number not equal to 1, and the number is positive. The calculator will handle any valid base, including fractions and decimals.
Tips & Common Mistakes
Tips
- Ensure the base is a positive number greater than 0 and not equal to 1, otherwise the logarithm is undefined.
- The number (argument) must be positive; logarithms of zero or negative numbers are not real.
- Use the step-by-step working to understand how the change-of-base formula is applied, which helps in learning.
- For quick mental checks, remember common logs: log base 10 of 100 is 2, log base 2 of 32 is 5.
Common Mistakes to Avoid
- Entering a base of 1, which is invalid because log base 1 is undefined.
- Entering a negative number or zero as the argument, which does not produce a real logarithm.
- Confusing the base and the argument, leading to incorrect results.
Last updated: August 13, 2026