Log Calculator (Natural Log, Common & Custom Base)

Log Calculator

Calculate natural logs, common logs, and custom base logarithms instantly.

Quick Select Base
log
(
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Enter a positive number (x > 0). Base (b) must be > 0 and ≠ 1.

Related Calculator: Scientific Calculator | Exponent Calculator

Logarithms are the foundation of advanced algebra, calculus, and computer science. Whether you are dealing with exponential decay or trying to decode binary algorithms, calculating logs by hand is nearly impossible.

Our free log calculator is designed to give you instant, mathematically flawless answers. Unlike standard handheld devices, this tool easily handles a common log (base 10), a natural log (base e), a log base 2 calculator, or any custom base you need.

What is a Logarithm?

At its core, a logarithm is simply the inverse (the opposite) of an exponent. It answers a very specific question: “How many of one number do we multiply to get another number?”

If we look at the equation:

23=8,weknowthat2×2×2=82^3 = 8, we\\know\\that\\2 \times 2 \times 2 = 8

If we write that exact same relationship as a logarithm, it looks like this:

log2⁡(8)=3\log_2(8) = 3

The small “2” is the base, the “8” is the argument (the number you want to reach), and the “3” is the exponent needed to get there. Our log solution calculator instantly finds that exponent for you, and automatically displays the equivalent exponential form so you can double-check the math.

Understanding the Different Bases

The letter “b” in the formula log_b(x) represents the base. While you can use any positive number as a base (except 1), there are three primary bases used in mathematics:

1. Common Log (Base 10)

If a math problem simply writes “log(x)” without a small number attached to the bottom, it is universally understood to be Base 10. Our calculator defaults to this automatically. Base 10 is used heavily in science to measure the Richter scale (earthquakes) and pH levels.

2. Natural Log (Base e)

Often written as ln(x), the natural logarithm uses Euler’s number (e = 2.718) as its base. If you need a natural log calculator, simply click the “Base e” button on our tool. This base is essential in physics, continuous compound interest in finance, and population growth models.

3. Binary Log (Base 2)

If you are studying computer science, you will frequently need a log 2 calculator. Because computers operate on binary code (0s and 1s), Base 2 logs are used to calculate data complexity, bit rates, and information theory formulas.

Frequently Asked Questions

If you are wondering how to use log on calculator devices (like a standard Casio or TI-84), you will notice they usually only have two buttons: [log] for base 10, and [ln] for base e. To calculate any other base, you must use the Change of Base Formula: divide the base-10 log of your number by the base-10 log of your target base. For example, to find log_2(8), you type log(8) / log(2). Our web tool automatically does this for you and shows the exact step-by-step formula!

A natural log calculator evaluates logarithms using Euler’s number (e = 2.718) as its base. In mathematics, this is written as ln(x) instead of log_e(x). Natural logs are incredibly important in advanced calculus, physics, and calculating continuous compound interest in finance. You can easily calculate these on our tool by selecting the “Natural (Base e)” quick-select button.

A log base 2 calculator (also known as a binary log) is used heavily in computer science, software engineering, and information theory because computers operate on a binary system of 0s and 1s. To use our log 2 calculator, simply click the “Binary (Base 2)” button, enter your value in the “x” box, and hit calculate to instantly find the exponent.

When you look at a math problem or use a calculator of log functions, you might see an equation written simply as log(100) with no small number at the bottom. In mathematics, an unwritten base is universally understood to be a Common Log, which is Base 10.

No, a log solution calculator cannot evaluate negative numbers or zero. Mathematically, the base of a logarithm must be a positive number greater than zero. Because you cannot raise a positive base to any real exponent and get a negative result, the number you are trying to evaluate (x) must always be greater than zero.