Log Calculator Online Free

A logarithm looks intimidating, but it asks a very simple question: to what power must a base be raised to get a given number? If 10 to the power 3 is 1,000, then the log base 10 of 1,000 is 3. That is all a logarithm is, and once that question is clear, the notation stops being frightening. This free log calculator answers that question for log base 10, for the natural log, and for any custom base you choose, in your browser, with nothing uploaded and no sign-up.

A log calculator is also called a logarithm calculator. It is the inverse of a power calculator: where a power calculator starts from the exponent and finds the result, a log calculator starts from the result and finds the exponent. The two tools undo each other, and using them together is one of the best ways to build intuition for both.

What It Is

The logarithm of a number, to a given base, is the exponent to which the base must be raised to produce that number. In symbols, if base^b equals x, then log base b of x equals b's exponent, written log_b(x). The number x is sometimes called the argument, and the base is the number being raised to a power. So log base 10 of 1,000 is 3, because 10^3 is 1,000. Log base 2 of 8 is 3, because 2^3 is 8. The natural log, written ln(x), is the logarithm to base e, where e is Euler's number, about 2.71828. So ln(e) is 1, because e^1 is e. Three bases dominate everyday use, and this calculator covers all three, plus any base you choose. Log base 10, also called the common logarithm, is the base used in science for orders of magnitude, in the pH scale, in decibels, and in the Richter scale for earthquakes. Log base 2, also called the binary logarithm, is the base used in computing, information theory, and anything that doubles or halves step by step. The natural log, base e, is the base used in calculus, in continuous growth and decay, in compound interest over very short intervals, and in many formulas in physics and statistics. A custom base covers everything else: log base 5, log base 7, or any base a particular problem requires. One restriction matters and is enforced honestly by the tool: logarithms are only defined for positive numbers. The log of zero is not defined, because no power of a positive base gives zero, and the log of a negative number has no ordinary real-number value, because a positive base raised to a real power is always positive. If you enter zero or a negative number, the tool shows a clear message instead of a meaningless result. Similarly, log base 1 is not defined, because 1 raised to any power is always 1, so it can never produce any other number; the tool refuses base 1 with a plain explanation.

How to Use It

Using the log calculator takes four short steps:

1. Enter the number. Type the positive number whose logarithm you want. For example, type 1000 for log base 10 of 1,000.
2. Enter a custom base if you need one. If you want a log to a base other than 10 or e, type that base in the custom base box. If you only want log base 10 or the natural log, you can leave it blank.
3. Choose what to show. Select all three kinds, log base 10 only, natural log only, or custom base only.
4. Click Calculate Logarithm. Read the result or results, each labelled with its base.

A practical tip: when you choose “All”, the tool also shows log base 2, because binary logarithms are so common in computing that seeing them alongside the others saves a second calculation. Another tip: results are rounded to a sensible display precision, so a logarithm that is not a whole number, such as ln(10), appears as 2.302585093, which is the value to ten significant figures, not a truncated guess.

Worked Examples

Worked examples are the fastest way to check a calculator, and these are the three live tests this tool is built around, plus everyday cases. Example 1: log base 10 of 1,000. Enter 1000 and select log base 10. The calculator gives 3. The working is the defining question: 10 to what power is 1,000? 10^1 is 10, 10^2 is 100, 10^3 is 1,000, so the answer is 3. This is also the reason log base 10 is used for orders of magnitude: a number with a log base 10 of 3 is in the thousands, one with 6 is in the millions, and one with -3 is in the thousandths. Example 2: natural log of e. Enter the value of e, about 2.718281828, and select natural log. The calculator gives 1. The working is the same question with base e: e to what power is e? e^1 is e, so the answer is 1. More generally, ln(10) is about 2.302585093, and ln(100) is about 4.605170186, which is exactly double, because 100 is 10 squared, and the logarithm of a square is twice the logarithm. Example 3: log base 2 of 8. Enter 8, enter custom base 2, and select custom base. The calculator gives 3. The working is 2^3 is 8, so the exponent is 3. This is the pattern behind computing: a file of 8 units doubling three times, a search that halves a list three times, or a binary tree three levels deep all involve log base 2. Example 4: A logarithm that is not a whole number. Enter 50 and select log base 10. The calculator gives about 1.69897. The working is that 10^1 is 10 and 10^2 is 100, so the log base 10 of 50 must lie between 1 and 2, closer to 2 because 50 is closer to 100 than to 10 on a multiplicative scale. Logarithms compress a huge range into a small one, which is why they are used for scales that span many orders of magnitude.

Why Logarithms Matter: Scales, Growth and the Laws of Logarithms

Logarithms matter because the world spans enormous ranges, and our senses and instruments respond to ratios rather than to raw differences. Sound is measured in decibels, which are logarithmic: an increase of 10 decibels means ten times the intensity, not ten units more. Acidity is measured in pH, which is a negative log base 10 of hydrogen ion concentration: a pH change of 1 means a tenfold change in acidity. Earthquakes are measured on a logarithmic scale: a magnitude 6 earthquake releases about 32 times the energy of a magnitude 5 earthquake. In each case, the logarithm turns multiplication into addition, which is far easier to read, compare, and plot. The laws of logarithms mirror the laws of exponents, because logarithms are exponents. The log of a product is the sum of the logs: log(100 × 10) is log(100) plus log(10). The log of a quotient is the difference of the logs. The log of a power brings the exponent down in front: log(10^3) is 3 × log(10), which is 3. The change-of-base formula lets any logarithm be computed from any other base: log base b of x equals ln(x) divided by ln(b), and that is exactly how this calculator computes a custom base, from natural logs, in your browser. These laws are worth learning because they turn hard multiplications into easy additions, and they explain why slide rules, which are physical logarithmic scales, could multiply before electronic calculators existed. Logarithms also describe growth and decay over time. Continuous compound interest uses the natural log, because e is the base of continuous growth: a quantity growing continuously at a rate r for time t is multiplied by e^(r × t), so the time needed to reach a target is found with a natural log. Radioactive decay, cooling, and the fading of a drug in the bloodstream all follow the same pattern, with a half-life that is a natural log divided by a decay constant. A log calculator therefore sits at the centre of finance, physics, biology, and statistics, not only in a mathematics classroom.

Limits: What This Tool Does and Does Not Do

Honesty about limits keeps a calculator trustworthy. First, this tool evaluates numeric logarithms only; it does not simplify logarithmic expressions, solve logarithmic equations, or show step-by-step symbolic working. The article explains the laws, but the tool itself returns numbers. Second, the argument must be positive and the base must be positive and not equal to 1, for the mathematical reasons given above; those inputs are refused with a clear message rather than an invented result. Third, results are displayed to ten significant figures, which is ample for everyday and classroom use, but a result that is an exact whole number, such as log base 10 of 1,000, is shown exactly as 3. Fourth, this tool does not convert between logarithmic scales such as decibels or pH directly; it gives the underlying logarithm, from which those scales follow by their own definitions.

Privacy: Browser Only

This calculator runs entirely in your browser. The number and base you type are processed on your own device, and nothing is uploaded to a server, stored in an account, or shared with anyone. There is no sign-up and no record kept of your calculations. Closing the page ends the session, and nothing about your numbers remains behind.

FAQs

What is a logarithm? A logarithm is the exponent to which a base must be raised to get a given number. Log base 10 of 1,000 is 3, because 10^3 is 1,000. Log base 2 of 8 is 3, because 2^3 is 8.

What is log base 10 of 1,000? Log base 10 of 1,000 is 3, because 10 to the power 3 is 1,000.

What is the natural log of e? The natural log of e is 1, because the natural log uses base e, and e^1 is e. Euler's number e is about 2.71828.

What is log base 2 of 8? Log base 2 of 8 is 3, because 2 to the power 3 is 8. Log base 2 is widely used in computing and information theory.

Why is there no log of zero or a negative number? No power of a positive base gives zero or a negative number, so no exponent answers the logarithm question for those arguments. That is why the tool refuses them with a clear message.

How do I change the base of a logarithm? Use the change-of-base formula: log base b of x equals ln(x) divided by ln(b). This calculator applies that formula for you when you enter a custom base.

Related Tools

Your Smart Kit offers a growing set of free online calculators for everyday mathematics, and several sit naturally alongside this log calculator. If you need the forward question, what a base to a power is, the exponent calculator answers it. If you need to add a sequence of terms, the sigma calculator sums them one by one. If you work with very large or very small numbers, the standard form calculator writes them as a coefficient times a power of ten. Each tool follows the same approach as this one: free, in your browser, with the working shown and the limits stated, so you can check a result rather than simply trusting it. A logarithm is only an exponent looked at from the other side. Ask what power gives the number, and the notation, the scales, and the growth formulas all fall into place.