Base 2 logarithm
log_2(8)3 because 2^3 = 8
Log Calculator
Use this free log calculator to find logarithms with any valid base, compare ln and log10 values, see change-of-base steps, check the matching exponent, and avoid invalid values such as zero, negative inputs, or base 1.
Custom basesln and log10Change-of-base stepsPower check
log_2(8)3
- ln(value)
- 2.0794415417
- log10(value)
- 0.903089987
- Power check
- 8
Steps
- Start with log base 2 of 8.
- Use the change-of-base formula: log_b(x) = ln(x) / ln(b).
- Substitute the values: ln(8) / ln(2) = 2.0794415417 / 0.6931471806.
- The answer is 3.
Recent answers
Recent log answers will appear here.
Input tips
The value must be greater than zero.
The base must be greater than zero and cannot equal 1. Enter e for a natural log base.
How to use the Log Calculator
- Enter the positive value you want to evaluate.
- Enter a positive base that is not 1, such as 2, 10, or e.
- Press Calculate log to see the logarithm, ln, log10, power check, and steps.
- Use examples, recent answers, or copy the answer while you study logarithms.
What people use it for
Calculate log base 2, base 10, natural log, or another custom base.
Check logarithm homework with change-of-base steps.
Compare log, ln, and log10 values from one input.
Confirm a logarithm by seeing the matching exponential power check.
See why zero, negative values, and base 1 are not valid real logarithm inputs.
Use the tab-only recent answer history while studying repeated log problems.
Quick examples
Common logarithm
log_10(1000)3 because 10^3 = 1000
Natural logarithm
ln(e^3)3 because e^3 gives the input value
Negative log answer
log_10(0.01)-2 because 10^-2 = 0.01
Custom base
log_5(625)4 because 5^4 = 625
Change of base
log_3(81)4 using ln(81) / ln(3)
Need the guide or a nearby tool?
Need a slower walkthrough, a related calculator, or the full library? These links keep you close to the task you started.
- How to use the Log CalculatorSee slower examples, input meanings, formula notes, result limits, and mistakes to avoid.
- CalculatorsBrowse related calculators and utilities in the same category.
- Calculator and utility guidesFind more guides that use real tool inputs and example numbers.
- All free toolsSearch every live tool, converter, calculator, AI helper, and browser utility.
Frequently asked questions
Quick answers about valid bases, ln, log10, change-of-base, exponential checks, and privacy.
What does a logarithm mean?
A logarithm asks which exponent makes the base turn into the value. For example, log_2(8) = 3 because 2^3 = 8.
What formula does the Log Calculator use?
It uses the change-of-base formula: log_b(x) = ln(x) / ln(b). This lets the calculator solve logarithms for any valid positive base except 1.
What do the main Log Calculator inputs mean?
The main inputs are the numbers, operation, mode, or known values the calculator needs. Keep units consistent, enter percentages the way the page label shows, and use the examples as a quick check before trusting the answer.
How should I read the Log Calculator answer?
Read the headline answer, then check the supporting lines and examples to understand how the calculator got there. If one input changes, rerun the tool and compare the new answer instead of guessing.
What should I double-check before trusting the Log Calculator?
Check units, signs, rounding, and the selected mode before copying the answer. If the number feels weird, rerun one of the examples first, then put your own values back in slowly.
What values can I enter?
The log value must be greater than zero. The base must also be greater than zero, and the base cannot be 1.
What is the difference between log, log10, and ln?
log10 means base 10, ln means base e, and a custom log lets you choose another base such as 2, 3, or 5.
Can I enter e as the base?
Yes. Type e in the base field to use the natural log base. The calculator also shows ln(value) automatically for every valid positive value.
How can I check a logarithm answer?
Rewrite it as an exponent. If log base b of x equals y, then b^y should equal x. The calculator shows this check in the result card.
Can logarithm answers be negative?
Yes. A logarithm can be negative when the value is between 0 and 1 for a base greater than 1, such as log_10(0.01) = -2.
Why can the log value not be zero or negative?
In real-number logarithms, no positive base can be raised to a real exponent and produce zero or a negative value. Use a complex-number tool if your class or project needs complex logs.
Why can the base not be 1?
A base of 1 never grows or shrinks. Since 1 raised to any power is still 1, log base 1 cannot tell you one clear exponent.
When should I use log10 instead of ln?
Use log10 when the problem is built around powers of 10, such as orders of magnitude. Use ln when the problem is built around base e, which is common in algebra, calculus, growth, and decay formulas.
Why is my answer a long decimal?
Many logarithms are irrational decimals. The calculator formats the result for practical checking, so use the exponent check when you need to see whether the rounded answer matches the original value.
Can this replace a symbolic algebra system?
No. This page is for real-number numeric logarithms and quick study checks. It does not simplify symbolic expressions, solve equations, graph functions, or return complex logarithms.
Is my log calculation history private?
Yes. Recent log answers stay only in the current browser tab while you use the page. They are not sent to a server.