Log Calculator

Introduction

A logarithm tells you what power you need to raise a base number to in order to get another number. For example, log base 2 of 8 equals 3, because 2 raised to the power of 3 equals 8. Logarithms are the opposite of exponents. They show up a lot in algebra, science, and everyday math problems. This log calculator lets you quickly solve logarithms with any base and any number. Just enter your values, and the calculator does the work for you. It's a simple way to check your homework or save time on tough problems.

How to Use Our Log Calculator

Enter a number and a base to calculate the logarithm. The calculator will give you the log value instantly.

Number (x): Type the number you want to find the logarithm of. This must be a positive number greater than zero.

Base (b): Enter the base of the logarithm. Common choices are 10 (common log), 2 (binary log), or e (natural log). The base must be a positive number and cannot equal 1.

Result: The calculator will show you the exponent that the base must be raised to in order to equal your number. In other words, if log_b(x) = y, the result is y.

What Is a Logarithm?

A logarithm answers one simple question: what power do you raise a base to in order to get a certain number? For example, log base 2 of 8 equals 3, because 2 raised to the 3rd power equals 8. In short, logarithms are the opposite of exponents.

How Logarithms Work

Every logarithm has three parts: the base, the argument (the number you're taking the log of), and the result. When you see log_b(x) = y, it means b^y = x. For instance, log₁₀(100) = 2 because 10² = 100.

Common Types of Logarithms

• Common logarithm (log): Uses base 10. This is the default on most calculators.
• Natural logarithm (ln): Uses base e (approximately 2.71828). It shows up often in science and advanced math.
• Binary logarithm (log₂): Uses base 2. It is widely used in computer science.

Key Logarithm Rules

There are a few important rules that make working with logarithms easier:

• Product Rule: log_b(x × y) = log_b(x) + log_b(y)
• Quotient Rule: log_b(x ÷ y) = log_b(x) − log_b(y)
• Power Rule: log_b(xⁿ) = n × log_b(x)
• Change of Base Formula: log_b(x) = log(x) ÷ log(b). This formula is especially handy when you need to convert between bases using a Fraction Calculator to simplify the resulting expression.

Important Things to Remember

You can only take the logarithm of a positive number. The base must also be positive and cannot equal 1. The log of 1 is always 0, no matter what the base is, because any number raised to the power of 0 equals 1.

Where Are Logarithms Used?

Logarithms are used in many real-world areas. The Richter scale for earthquakes, decibels for sound, and pH levels in chemistry all rely on logarithms. You can explore pH calculations further with our pH Calculator. They help us work with very large or very small numbers by turning multiplication into addition, which makes calculations much simpler. Logarithms also play a key role in solving algebraic equations — for example, when working with quadratic equations that involve exponential terms. If you need to analyze the rate of change of a logarithmic function, or calculate the slope of a log curve at a given point, those tools pair well with this calculator. You might also find logarithms useful when computing percent change on exponential growth data or when working with standard deviations of log-transformed datasets. For more advanced work, our Derivative Calculator can help you find the derivative of logarithmic functions, and the Percentage Calculator is handy for converting log-based results into percentages.

Frequently Asked Questions

What is log base 10 of 1000?

Log base 10 of 1000 equals 3. This is because 10 raised to the power of 3 equals 1000 (10 × 10 × 10 = 1000).

Can I find the base if I know the argument and the result?

Yes. Leave the base field empty and enter the argument (x) and the result (y). The calculator will solve for the base using the formula b = x^(1/y).

Can I find the argument if I know the base and the result?

Yes. Leave the argument field empty and type in the base and the result. The calculator uses the formula x = b^y to find the missing argument.

Why can't the base equal 1?

Because 1 raised to any power is always 1. That means log base 1 cannot produce different results for different arguments. The math simply does not work, so base 1 is not allowed.

Why can't I take the log of a negative number?

Logarithms are only defined for positive numbers. There is no real exponent you can raise a positive base to and get a negative result. So the argument (x) must be greater than zero.

What is the difference between log and ln?

Log usually means log base 10, also called the common logarithm. Ln means the natural logarithm, which uses base e (about 2.718). Both work the same way, just with different bases.

How do I type Euler's number (e) as the base?

Simply type the letter e in the base field. The calculator will treat it as Euler's number, which is approximately 2.71828. You can also click the dropdown arrow and select e from the list.

Does this calculator support scientific notation?

Yes. You can type numbers in scientific notation like 5e3 (which means 5,000) or 2.5e-4 (which means 0.00025) into any field.

What does the graph tab show?

The graph tab plots the logarithmic function y = log_b(x) for your chosen base. You can change the base and the maximum x value to see how the curve shape changes.

Why does log of 1 always equal 0?

Any number raised to the power of 0 equals 1. So no matter what base you use, log_b(1) = 0. For example, 10⁰ = 1, 2⁰ = 1, and e⁰ = 1.

What is the change of base formula?

The change of base formula lets you convert a logarithm to a different base. It says log_b(x) = log(x) ÷ log(b). You can use log base 10 or ln for both parts. This calculator uses this formula behind the scenes.

Can I use this calculator for log base 2?

Yes. Type 2 in the base field or select it from the dropdown menu. Log base 2 is common in computer science and is sometimes called the binary logarithm.

What happens if I fill in all three fields?

The calculator needs exactly one empty field to know what to solve for. If all three fields are filled in, it will ask you to clear one field before calculating.

How accurate are the results?

Results are shown up to 8 decimal places. The calculator uses your browser's built-in math functions, which provide high precision for most everyday and school-level problems.

How do I copy my result?

Click the copy icon next to the result field. The value will be copied to your clipboard, and a small "Copied!" message will appear to confirm.