Calculate logarithms with Base e, Base 10, Base 2, or Custom.
Mathematics is a language that describes the universe, and logarithms are one of its most powerful tools. Whether you are a student grappling with algebra, a computer scientist analyzing algorithms, or an engineer calculating sound intensity, understanding logarithms is essential. This guide dives deep into the concept of logarithms, exploring their definition, properties, common bases, and real-world applications. Our Log Calculator above makes solving these problems instant, but understanding the "why" and "how" behind the numbers will give you a true mastery of the subject.
At its core, a logarithm is the inverse operation of exponentiation. Just as subtraction is the opposite of addition and division is the opposite of multiplication, logarithms "undo" exponents. A logarithm answers a very specific question: "To what power must we raise a specific base to produce a certain number?"
Here is a breakdown of the components:
Example: Why is log10(1000) = 3? Because 103 = 1000. The base (10) multiplied by itself 3 times equals 1000.
While you can have a logarithm with any valid base (like 5, 12.5, or 100), three specific bases appear so frequently in science and mathematics that they have special names and notations.
The common logarithm is written simply as log(x) (without a subscript). If you see "log" on a standard calculator with no base specified, it is almost certainly Base 10.
The natural logarithm uses the mathematical constant e (Euler's number), which is approximately equal to 2.71828. This is arguably the most important logarithm in higher mathematics.
The binary logarithm is the language of computers.
Logarithms follow a set of algebraic rules that make simplifying complex equations much easier. These rules essentially turn multiplication into addition and division into subtraction.
| Rule Name | Formula | Explanation |
|---|---|---|
| Product Rule | logb(M · N) = logb(M) + logb(N) | The log of a product is the sum of the logs. |
| Quotient Rule | logb(M / N) = logb(M) - logb(N) | The log of a quotient is the difference of the logs. |
| Power Rule | logb(Mk) = k · logb(M) | The exponent of the argument can be moved to the front as a multiplier. |
| Zero Rule | logb(1) = 0 | Any base raised to the power of 0 equals 1. |
| Identity Rule | logb(b) = 1 | Any base raised to the power of 1 is itself. |
Most physical calculators and standard computer functions only calculate log (Base 10) and ln (Base e). What do you do if you need to calculate log5(100)?
You use the Change of Base Formula. This formula allows you to rewrite a logarithm of any base in terms of a base your calculator supports (usually 10 or e).
Example Calculation: Calculate log5(100).
Our Log Calculator tool automatically applies this formula when you select the "Custom" base option.
Logarithms are not just abstract math concepts; they are embedded in the fabric of how we measure the world. Here are common examples where logs are indispensable:
The Richter scale is logarithmic (base 10). This means an earthquake with a magnitude of 6.0 is not just "one unit" stronger than a 5.0. It has a shaking amplitude that is 10 times greater. A magnitude 7.0 earthquake is 100 times stronger than a 5.0. This logarithmic scale allows scientists to represent huge differences in energy on a simple 1-10 scale.
Human hearing is incredibly sensitive. We can hear a pin drop and a jet engine, even though the jet engine is trillions of times more powerful. To manage this huge range, sound intensity is measured in decibels (dB), which is a logarithmic unit. An increase of 10 dB represents a ten-fold increase in sound intensity. 20 dB is 100 times more intense than 0 dB.
In chemistry, pH measures the acidity or alkalinity of a solution. The formula for pH is: pH = -log10[H+], where [H+] is the concentration of hydrogen ions. Because of the negative log, a low pH means high acidity (lots of H+ ions), and a change of one pH unit represents a 10-fold change in ion concentration.
If you want to know how long it will take for your investment to double, you use logarithms. The formula to solve for time (t) in compound interest often requires taking the natural logarithm of both sides of the equation. This is the math behind the famous "Rule of 72".
Logarithms were invented in the early 17th century by the Scottish mathematician John Napier (published in 1614). Before calculators and computers, multiplying large numbers was tedious and error-prone. Napier realized that by using logarithms, multiplication could be turned into addition, which is much faster.
For over 300 years, "log tables" and slide rules (which are mechanical analog computers based on logarithmic scales) were the primary tools for scientists, engineers, and navigators. They were used to design bridges, navigate ships across oceans, and even send rockets to the moon. Today, while digital calculators do the heavy lifting, the concept remains vital for mathematical modeling.