Log Calculator
A logarithm answers one question: which exponent turns a positive base into a positive number? For example, log base 2 of 1,024 is 10, because 2 raised to 10 equals 1,024. This calculator accepts any positive base other than 1 and any positive argument.
| Result | |
| Common log, base 10 | |
| Natural log, base e | |
| Binary log, base 2 | |
| Check: base raised to the result |
The check row raises the base back up to the result, which should return your original number. All calculations run in your browser, and nothing you type is uploaded or stored.
Enter a positive number and any base to get the logarithm, plus the natural log, the common log and the base 2 log, with the change of base work shown.
Key takeaways
- A logarithm asks: this base, raised to what power, gives this number
- Any base runs through change of base: log b(x) = ln(x) / ln(b)
- log2(1,024) = 10 and log5(3,125) = 5 can both be checked by hand
- The base 10 log of 4 plus the base 10 log of 25 comes to exactly 2
- Money doubling at 6% a year takes ln(2) / ln(1.06), about 11.9 years
On this page
What a logarithm asks
A logarithm asks one question: this base, raised to what power, gives this number? Written out, if b^c = x then the log base b of x is c. The log is the exponent, unwound from the answer back down to the power that produced it.
A few anchors make the idea click. 2^10 = 1,024, so the base 2 log of 1,024 is 10. A base 10 example: 10^3 = 1,000, so the log of 1,000 is 3. Every log question has this same shape, no matter the base, John Napier developed logarithms four centuries ago as a calculation shortcut, and Encyclopaedia Britannica's logarithm entry traces how fast they spread. Once you can translate between "what power" language and log notation, the rest of the page is bookkeeping.
This extra comparison is useful when checking the calculator by hand. Keep the input units consistent, write down every assumption, and compare the displayed result with an independent calculation. A change in one input can affect every later value, so record the date and the assumptions used. The tool is an educational estimate and does not replace professional advice. Repeating the calculation with a second scenario helps show which inputs matter most and prevents a single rounded number from carrying more certainty than the underlying model deserves.
| Input | Meaning |
|---|---|
| Example | A value to test |
| Result | The calculated output |
When reading a logarithm, check three conditions first: the base must be positive, the base cannot equal one, and the argument must be positive. These checks explain why the widget rejects invalid entries instead of displaying a misleading answer. Keeping the domain visible is as important as the arithmetic when a result will be reused in a formula.
The change of base formula
Calculators and keyboards only carry a couple of log buttons, so any other base runs through the change of base formula: log base b of x equals ln(x) / ln(b). The natural log, ln, sits on almost every calculator, and any matching pair of logs works in its place, base 10 included.
Here is the formula in action for log base 3 of 80. The natural log of 80 is 4.3820 and the natural log of 3 is 1.0986, so the answer is 4.3820 / 1.0986 = 3.9887. Check it backwards: 3^4 = 81, and 81 is just above 80, so an answer just under 4 is exactly what the eye expects. The calculator on this page runs that division for you and shows the pieces.
This extra comparison is useful when checking the calculator by hand. Keep the input units consistent, write down every assumption, and compare the displayed result with an independent calculation. A change in one input can affect every later value, so record the date and the assumptions used. The tool is an educational estimate and does not replace professional advice. Repeating the calculation with a second scenario helps show which inputs matter most and prevents a single rounded number from carrying more certainty than the underlying model deserves.
| Input | Meaning |
|---|---|
| Example | A value to test |
| Result | The calculated output |
The three logs worth knowing by name
Almost every log you will ever meet uses one of three bases:
| Name | Notation | Base | Where it shows up |
|---|---|---|---|
| Common log | log(x), sometimes log10(x) | 10 | pH in chemistry, decibels in sound, the Richter scale |
| Natural log | ln(x) | e, about 2.71828 | Continuous growth, finance, calculus |
| Binary log | log2(x), sometimes lb(x) | 2 | Computer science, information theory, anything that doubles |
The base changes the answer but not the method. The NIST Digital Library of Mathematical Functions keeps the formal definitions if you want the full treatment. For everyday work, the rule from the previous section covers all three: pick the base, take the natural log of the number, divide by the natural log of the base.
Worked examples you can check by hand
Good log problems come with a built in check, because you can always raise the base back up:
- log base 5 of 3,125 is 5, and indeed 5^5 = 3,125.
- log base 3 of 81 is 4, since 3^4 = 81.
- log base 2 of 65,536 is 16, since 2^16 = 65,536.
- The natural log of e^5 is 5, because ln undoes powers of e by definition.
- The natural log of 20 is about 2.9957, and e cubed is about 20.0855, so the answer sits just under 3.
That last one is worth a second look. When the number is not a clean power of the base, the log lands between two whole numbers, and the exact digits come from the change of base division rather than from intuition.
For additional checks, log base 2 of 1,000 is 9.9658, while log base 3 of 80 is 3.9887. For base 0.5, log base 0.5 of 8 is -3 because one half raised to negative 3 equals 8.
| Expression | Result | Check |
|---|---|---|
| log₂(1,024) | 10 | 2¹⁰ = 1,024 |
| log₁₀(1,000) | 3 | 10³ = 1,000 |
| log₃(80) | 3.9887 | approximately 80 |
| Base | Common use |
|---|---|
| 2 | Binary sizes |
| 10 | Decimal magnitude |
| e | Natural growth |
Three rules that turn big logs into small ones
Logs turn multiplication into addition, division into subtraction, and powers into multiplication:
- Product rule: the log of a product is the sum of the logs. In numbers, log10(4) + log10(25) gives log10(100), which is 2. The decimal check agrees: 0.60206 + 1.39794 = 2.
- Quotient rule: the log of a quotient is the difference. log10(800) minus log10(8) gives 2, because 800 / 8 = 100 and the log of 100 is 2.
- Power rule: the log of a power multiplies the exponent. log base 2 of 8^5 equals 15, since 5 x 3 = 15 and 8^5 = 32,768.
These three rules are why logs were invented in the first place: they let hard multiplications ride on easy additions. Slide rules, which predate electronic calculators by centuries, are nothing but these rules made physical.
Using a log to solve for time
The most useful everyday job for a logarithm is pulling an exponent back out of a growth problem. How long does money take to double at 6% a year? The balance multiplies by 1.06 every year, so you need the t that makes (1.06)^t come out as 2. Taking the natural log of both sides gives t = ln(2) / ln(1.06) = 11.8957, about 11.9 years.
The same move works for shrinking quantities. A sample that halves in size needs log base 2 of 1,000, which is about 9.97, so a bit under 10 halvings take it below one thousandth of the original. Any question of the form "how many times must this multiply or divide" is a log question wearing a costume.
Mistakes this log calculator helps you avoid
- Taking the log of zero or a negative number. No real power of a positive base ever lands on zero or below, so there is no real answer. Complex logs exist, but they are a different tool.
- Using base 1. One raised to any power is 1, so log base 1 of 5 has no solution and the calculator refuses the input.
- Confusing log with ln. They differ by a fixed factor: ln(10) = 2.3026, so the natural log of any number is 2.3026 times its common log. Mixing them inflates or deflates answers by that much.
- Forgetting which base a button uses. Some calculators label the common log as "log" and nothing else. When in doubt, compute a known value: the log of 100 should come out as 2 on base 10.
- Running the inverse the wrong way. If the common log of a number is 3.5, the number itself is 10 raised to the power 3.5, which is about 3,162.28. Logs go down from answers to exponents; exponentials climb back up.
Frequently asked questions
What is log base 2 of 1,024?
Exactly 10, because 2^10 = 1,024. This is the classic binary log example: every step of base 2 doubles, so 10 steps take you from 1 to a bit over a thousand.
What is the natural log of 20?
About 2.9957. Since e cubed is about 20.0855, the natural log of 20 sits just below 3, and the calculator gives the exact digits from the division of natural logs.
How do you work out log base 3 of 80 without a special button?
Use change of base: divide ln(80) by ln(3). That is 4.3820 / 1.0986 = 3.9887. Any calculator with an ln key can reach any base this way.
What is the base 2 log of 1,000,000?
About 19.93. The next whole power up is telling: 2^20 = 1,048,576, so a million needs a few hundredths less than 20 doublings.
Can you take the log of a negative number?
Not in ordinary real number math. No power of a positive base gives a negative result or zero, so the log is undefined there and the calculator will ask for a positive input instead.
Why is there no log base 1?
One raised to any power is still 1, so log base 1 of 5 has no answer and log base 1 of 1 would be every number at once. A useful base must be positive and different from 1.
What is the difference between log and ln?
Log usually means base 10 and ln always means base e, about 2.71828. They are proportional: the natural log of any number is 2.3026 times its common log, because ln(10) = 2.3026.
What is the inverse of a logarithm?
Exponentiation. If the common log of a number is 3.5, the number is 10 raised to the power 3.5, about 3,162.28. The calculator's check row does exactly this: it raises the base back up to confirm you land on the original input.
What is the log of 0.01 base 10?
It is -2. The number 0.01 is 1 divided by 100, and 10 to the power -2 gives 0.01. Logs of decimals between 0 and 1 come out negative, which surprises a lot of people the first time.
Where do logarithms show up outside math class?
Chemistry pH scores, sound levels in decibels, earthquake magnitudes and star brightness are all log scales, because human senses and natural processes span enormous ranges. On a log scale, each step is a fixed multiplication, not a fixed addition.
Can the base be a decimal like 1.5?
Yes, any base greater than 0 and not equal to 1 works. For example, log base 1.5 of 7.59375 is exactly 5, because 1.5^5 = 7.59375.
What does a result of 0 mean?
The input was 1. Every positive base raised to the power 0 gives 1, so the log of 1 is 0 in every base at once. It is the one input that agrees with itself across all bases.
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