Interactive Logarithm Calculator
Solve logarithm problems, visualize the relationship with exponents, and explore how they work with our interactive tool.
Logarithm Solver
Solves equations in the form: logb(y) = x. Leave one field blank to solve.
Interactive Visualizer
See the relationship between logb(y) = x and y = bx. Adjust the sliders to explore!
Change of Base Calculator
Use this to calculate a logarithm with any base using a new, more common base (like 10 or 'e').
Logarithm Cheat Sheet & Concept Primer
The Core Relationship
A logarithm is the inverse of an exponent. It answers the question: "what exponent gets me this number?"
Log Form: logb(y) = x
Exponent Form: bx = y
Special Logarithms
Common Log: A log with base 10. Written as log(y).
Natural Log: A log with base e (โ2.718). Written as ln(y).
Product Rule
The log of a product is the sum of the logs of its factors.
logb(m * n) = logb(m) + logb(n)
Quotient Rule
The log of a quotient is the log of the numerator minus the log of the denominator.
logb(m / n) = logb(m) - logb(n)
Power Rule
The log of a number with an exponent is the exponent times the log of the number.
logb(mp) = p * logb(m)
Change of Base Rule
Allows you to convert a log of any base to a base you can work with (like 10 or e).
logb(y) = logc(y) / logc(b)
Types of Logarithms in Detail
While a logarithm can have any positive number as its base, a few are so common they have special names and applications.
Common Logarithm (Base 10)
Written as log(x), the common log uses base 10. Before calculators, it was the foundation of slide rules and log tables. It's still used in many scientific scales where values span many orders of magnitude.
- pH Scale: Measures acidity in chemistry.
- Decibel Scale: Measures sound intensity.
- Richter Scale: Measures earthquake magnitude.
Natural Logarithm (Base e)
Written as ln(x), the natural log uses Euler's number (e โ 2.718) as its base. It arises naturally in calculus and describes processes of continuous growth or decay, making it essential in finance, physics, and biology.
Binary Logarithm (Base 2)
Written as log2(x), the binary log uses base 2. It is fundamental in computer science and information theory, as computers operate in a binary (base-2) system. It often answers questions like, "How many bits are needed to represent a number?"
How to Use the Log Calculator
1. Enter Two Values
In the equation logb(y) = x, fill in any two of the three fields: the base (b), the argument (y), or the result (x).
2. Click Calculate
The calculator will solve for the unknown variable you left blank, using the fundamental properties of logarithms.
3. Review the Solution
See the answer, the formula used, and a step-by-step explanation of how the result was calculated, including the conversion to exponential form.
Understanding Logarithms
What is a Logarithm?
A logarithm answers the question: "What exponent do I need to raise a certain base to, to get another number?" In logb(y) = x, 'x' is the exponent you raise 'b' to in order to get 'y'. The equivalent exponential form is bx = y.
What is a Common Log?
A "common log" is a logarithm with base 10. It's often written as log(y), without a visible base. It answers the question, "10 to what power equals y?"
What is a Natural Log (ln)?
A "natural log" is a logarithm with base e (Euler's number, approximately 2.718). It's written as ln(y). Natural logs are widely used in science, finance, and mathematics.
Change of Base Formula
To find a log with a base your calculator doesn't have, you can use the change of base formula: logb(y) = logc(y) / logc(b). You can use any new base 'c', but 10 or e are most common. For example, log3(9) = log(9) / log(3) = 2.
A Brief History of Logarithms
Before electronic calculators, logarithms were one of the most important mathematical tools for scientists, engineers, and astronomers, turning complex multiplications into simple additions.
The Invention by John Napier (1614)
Logarithms were first introduced by Scottish mathematician John Napier. His goal was to simplify the incredibly tedious calculations required in astronomy and navigation. His work allowed for multiplication and division to be performed by addition and subtraction, a revolutionary concept at the time.
Common Logarithms and Henry Briggs
Shortly after Napier's publication, English mathematician Henry Briggs collaborated with him to refine the idea. They developed "common logarithms," which use base 10. This innovation made logarithms far more practical for everyday calculations, and Briggs spent much of his life creating detailed log tables.
The Age of the Slide Rule
The invention of logarithms directly led to the creation of the slide rule, an analog computer that was the essential tool for engineers and scientists for over 300 years. By using logarithmic scales, slide rules enabled rapid calculations, a role they held until the electronic calculator became widespread in the 1970s.