Log Calculator
This log calculator finds the logarithm of a number in any base, plus its natural log (ln) and common log (log base 10). Enter the number and the base and the log calculator returns the result, so it works as a logarithm calculator, a log base 2 calculator, and a log and ln calculator in one.
Results
The Log Calculator is part of our Algebra collection. Browse more algebra to size, convert, and check your work alongside this tool.
What the log calculator shows
A logarithm answers the question: to what power must the base be raised to give this number? log base 10 of 1,000 is 3 because 10³ = 1,000. The log calculator works in any base and also returns the natural log, which uses the base e ≈ 2.71828, and the common log, base 10.
How logarithms are calculated
This change-of-base rule lets any logarithm be found from natural logs. The basic rules also help: log(ab) = log a + log b, log(a ÷ b) = log a − log b, and log(an) = n log a.
Log calculator examples
log base 10 of 72:
log base 2 of 10 is 3.321928, and log base 10 of 1,000 is exactly 3.
How to use the logarithm calculator
Enter the number and the base. The logarithm calculator returns the log in that base, the natural log, and the common log. Use base 2 for computing and information theory, base 10 for decibels and pH, and base e for growth and decay. The number must be greater than zero, and the base must be positive and not 1.
Reading the log calculator result in base 2
A log of 1 is always 0 in any base, and logs of numbers between 0 and 1 are negative. Each step of 1 in a base-10 log means ten times larger. In base 2, each step means double, which is why log base 2 of 1,024 is 10.
When to use the logarithmic calculator
- Checking homework with the logarithmic calculator
- Converting between log and ln
- Finding bits needed with a binary logarithm calculator
- Working with decibels, pH, or growth rates
Common log calculator mistakes
- log(a + b) is not log a + log b.
- Using base 1 or a negative base.
- Confusing ln with log base 10.
Tips for accurate results
- Check the base before reading the answer.
- Use log rules to simplify before calculating.
- Remember log of 1 is 0 in any base.
Natural log solver and log to ln
Natural logs use e, about 2.71828, and appear wherever growth is continuous, from interest to population to radioactive decay. To move between the two common bases, multiply a base-10 log by 2.302585 to get ln, or divide ln by the same number to get log base 10. A natural log solver and a log to ln calculator are the same tool viewed from either side. The change-of-base rule also shows that loga b and logb a are reciprocals, and the fraction calculator handles the fraction arithmetic that often follows.
Binary logarithm calculator uses
A binary logarithm tells you how many times to halve or double. It gives the number of bits needed for a range of values and the number of steps in a binary search.
Logarithms reference
| x | log10 x | ln x | log2 x |
|---|---|---|---|
| 2 | 0.3010 | 0.6931 | 1 |
| 10 | 1 | 2.3026 | 3.3219 |
| 72 | 1.8573 | 4.2767 | 6.1699 |
| 1,000 | 3 | 6.9078 | 9.9658 |
Log Calculator FAQ
What is a logarithm?
The power a base must be raised to in order to get a number: log_b x = y means b^y = x.
How do I convert log to ln?
ln x = log10 x × 2.302585.
Can you take the log of zero or a negative number?
No, not in real numbers. The number must be greater than zero.
How do you calculate a log without a calculator?
Split the number into a value between 1 and 10 times a power of 10, then add known logs. log10(2) is about 0.301, log10(3) about 0.477, and log10(7) about 0.845. For example, log10(480) = log10(4.8) + 2, and log10(4.8) = log10(16 × 3 ÷ 10) is about 0.681, so the answer is about 2.681.