Logarithm Calculator
Evaluate any logarithm instantly, log base b, natural log (ln) and common log (log₁₀). Fully usable on mobile.
Convert between logarithmic form logb(x) = y and exponential form by = x. Enter any two values to find the third.
Apply the change-of-base formula to rewrite any logarithm in terms of log₁₀ or ln. Useful when your calculator only has log₁₀ and ln buttons.
Place the unknown x in any position, in the argument, the base, or the result. The solver finds x algebraically and shows every step.
Enter a multi-term log expression and condense it into a single logarithm using the Product, Quotient and Power rules. Use + for product, - for quotient, and a leading coefficient for power. Example: 2*log(a)+log(b)-log(c)
Find the derivative d/dx of a logarithmic function. Enter the function and base. Supports ln(x), log(x), log_b(x), and functions of x inside the log.
How to Use the Logarithm Calculator
From choosing a base to a full answer in seconds. Fully usable on mobile.
Every calculation runs inside your own browser. No account, no tracking, nothing ever uploaded.
Logarithm Calculator, Free and Online
A logarithm calculator finds the power a base must be raised to in order to produce a given number, and shows the answer in both logarithmic and exponential form. This one supports any base, converts between logarithmic and exponential notation, applies the change of base formula, and runs entirely inside your browser.
Most logarithm problems only need one of a handful of operations, but finding the right one quickly matters. This free logarithm calculator tool covers them all in one place. Evaluate any logarithm with a common preset, log base 10, ln, or log base 2, or type in your own custom base. Convert freely between logarithmic form and exponential form, or apply the change of base formula when your own calculator only has log and ln buttons. Pro Mode adds a solve for x tool that places the unknown in the argument, the base or the result, an expression condenser that combines multi term log expressions using the product, quotient and power rules, and a derivative calculator that differentiates any logarithmic function step by step.
A logarithm and an exponential are two ways of writing the same relationship. Reading log base b of x equals y is the same as saying b raised to the power y equals x, and this calculator moves freely between both forms.
1Simple vs Pro, What Each Mode Includes
| Feature | Simple Mode | Pro Mode (Free) |
|---|---|---|
| Evaluate log base b of x, any base | Included | Included |
| Log to exponential conversion, both directions | Included | Included |
| Change of base formula | Included | Included |
| Solve for x (in the argument, base or result) | Not included | Included |
| Expression condenser (product, quotient, power rules) | Not included | Included |
| Derivative calculator, full step by step | Not included | Included |
| Step by step working, calculation history, law reference | Not included | Included |
2Common Bases, Quick Reference
The mistake that trips up the most students is forgetting that a logarithm's argument must always be positive. A logarithm answers the question of what power the base needs in order to produce a given number, and since a positive base raised to any real power can never produce a zero or negative result, the argument can never be zero or negative either. The base has its own restriction too, it must be greater than zero and cannot equal 1, since 1 raised to any power always stays 1 and could never uniquely produce any other number. This calculator checks both conditions automatically and tells you immediately if either one is violated.
3Where the Logarithm Actually Came From
Every log button on this page traces back to a single 1614 publication. According to the Smithsonian's National Museum of American History, the Scottish landowner John Napier spent nearly twenty years developing a way to turn multiplication into simple addition, publishing his results in Mirifici Logarithmorum Canonis Descriptio in 1614. Before logarithms, an astronomer computing a planetary orbit could spend hours multiplying large numbers by hand. Napier's tables let a calculator look up two logarithms, add them, and read the product straight off the same table. The French mathematician Pierre-Simon Laplace later said that by shortening the labor involved, logarithms doubled the life of the astronomer, and Johannes Kepler used Napier's method directly in the calculations that fed into Newton's theory of gravitation.
| Question | Quick Answer |
|---|---|
| Who invented logarithms? | John Napier |
| When were they published? | 1614 |
| What problem did they solve? | Turning multiplication into addition |
| Who praised their impact centuries later? | Pierre-Simon Laplace |
Every calculation runs inside your own browser using JavaScript, nothing is uploaded or sent to a server, and no account or signup is required.
4Frequently Asked Questions
Grounded in the Smithsonian's own account of John Napier's invention of logarithms, not the assumption that logarithms have always existed as a calculator function. The full account is available on the National Museum of American History's website. For related tools on this site, see the Scientific Calculator and the full SmallStudyTools.com tool library.
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