Quick answer
Enter the value, enter the base, and press Calculate log. For example, log_2(8) returns 3 because 2^3 = 8. The calculator also shows ln(value), log10(value), the change-of-base formula, and a power check.
The value must be greater than 0. The base must be greater than 0 and cannot equal 1. Type e in the base field when you want a natural-log base.
Quick start
- Enter the value you want inside the logarithm, such as 8, 1000, or 0.01.
- Enter the base, such as 2, 10, 5, or e.
- Press Calculate log.
- Review the main answer, ln(value), log10(value), power check, and steps.
- Use the examples, recent answers, or Copy answer while comparing log problems.
What the inputs mean
The value is the number you are trying to reach. In log_2(8), the value is 8. The base is the number being raised to a power. In log_2(8), the base is 2.
The answer is the exponent. If the calculator returns 3, it means the base raised to the third power should match the value. That is why log_2(8) = 3 and 2^3 = 8 say the same thing in two different forms.
This inverse relationship is the main idea to keep in your head. Exponents move from a base to a value. Logarithms ask which exponent did that job.
Valid values and bases
Real-number logarithms have strict input rules. The value must be greater than zero because no positive base raised to a real exponent becomes zero or a negative number.
The base must also be greater than zero, and it cannot be 1. A base of 1 never grows or shrinks because 1 raised to any power is still 1, so it cannot point to one clear exponent.
A logarithm answer can be positive, zero, or negative. For a base greater than 1, values above 1 create positive answers, value 1 creates answer 0, and values between 0 and 1 create negative answers.
Change of base
The calculator uses log_b(x) = ln(x) / ln(b). This lets it calculate logarithms for any valid base, even when the browser only gives direct natural-log and common-log math.
The change-of-base step is also useful for checking homework. If your class asks for log_3(81), the calculator can show that it is the same as ln(81) / ln(3), which returns 4 because 3^4 = 81.
Common log, natural log, and custom base
log10 means base 10. It is useful for powers of 10, orders of magnitude, scientific notation comparisons, pH-style examples, decibels, and scale problems.
ln means natural log, which uses base e. Natural logs show up often in algebra, calculus, growth, decay, compound interest models, and exponential formulas.
A custom base lets you ask a more specific exponent question. Base 2 is common in computer science. Base 3, 5, or another positive base can appear in algebra practice, repeated-growth examples, or problem-specific models.
How to check the result
The fastest check is to turn the logarithm back into an exponent. If log_b(x) = y, then b^y should equal x. For log_10(1000), the answer is 3 because 10^3 = 1000.
Rounded decimal answers may not reproduce the value perfectly when you type them back into another tool. That is normal. Use the exponent check as a reasonableness test, and keep extra decimals if your class, spreadsheet, or project needs a tighter tolerance.
Examples
Examples from the log calculator
3 because 2^3 = 8
3 because 10^3 = 1000
3 because e^3 gives the input value
-2 because 10^-2 = 0.01
4 because 5^4 = 625
4 using ln(81) / ln(3)
Common mistakes to avoid
- Do not put 0 or a negative number inside the log when you need a real-number answer.
- Do not use base 1. It cannot identify one clear exponent.
- Do not confuse log10 with ln. log10 uses base 10; ln uses base e.
- Do not assume a negative log answer means something went wrong. log_10(0.01) = -2.
- Do not round too early when the next step depends on the log answer.
- Do not treat this numeric calculator as a symbolic algebra or graphing system.
When not to rely on it alone
Use the Log Calculator for real-number numeric logarithms, quick study checks, input validation, and copied results. It is not meant to simplify symbolic expressions, solve logarithmic equations, graph log functions, or return complex logarithms.
If your class, lab, spreadsheet, or programming language has a required rounding rule, use that rule. If your problem involves negative log inputs, branch cuts, or complex values, use a complex-number tool or follow your course notes.
For nearby checks, try the Exponent Calculatorto reverse a log answer, the Scientific Notation Calculatorfor powers of 10, or the Root Calculator when the exponent question turns into a root.
FAQs from the log calculator
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.
Research and references
These references shaped the logarithm definition, change-of-base explanation, inverse exponent relationship, and real-number guardrails used in this guide.
History, privacy, and copying
Recent log answers stay visible in the page while you work. The history is kept only in the current browser tab and is not sent to a server.
Copy answer copies the expression and result so you can paste it into notes, homework, a message, or another document.