Logarithm Calculator

The logarithm function is a mathematical function that represents the inverse operation of exponentiation. It is used to solve equations involving exponential growth or decay and to convert between exponential and logarithmic forms.
log10(1)=  0log_{10}(1)=\;00
Photo of calculator author Wes NolteCreated by Wes Nolte.Last updated Apr 25, 2024.

The Logarithm Calculator Formula

Calculating logarithms involves finding the exponent to which a specified base must be raised to obtain a given number. Here is the formula and the general steps to calculate a logarithm.

y=logb(x)y = log_{b}(x)

Step 1: Identify the Base and the ArgumentThe logarithm function has a base, which is the number that you're taking the logarithm of, and an argument, which is the number for which you're finding the logarithm. For example, in the logarithm expression log⁡b(x)log_b(x), 'b' is the base and 'x' is the argument.Step 2: Use a Calculator (for Common Bases)Most calculators have buttons for common logarithms (base 10) denoted as 'log' and natural logarithms (base 'e') denoted as 'ln'. To calculate log⁡10(x)log_{10}(x) (logarithm base 10) or ln⁡(x)ln(x) (natural logarithm), enter the number 'x' and press the respective button.Step 3: Using Change of Base Formula (for Different Bases)If you want to calculate a logarithm with a base other than 10 or 'e', you can use the change of base formula:
log⁡b(x)=log⁡c(x)log⁡c(b)log_b(x) = rac{log_c(x)}{log_c(b)}
Here, 'b' is the base of the logarithm you want to calculate, and 'c' is any positive number, usually 10 for common logarithms or 'e' for natural logarithms. Use a calculator to find log⁡c(x)log_c(x) and log⁡c(b)log_c(b), then divide the logarithms to get the result.

Definition of the Logarithm Function

The logarithm function is typically denoted as "log" and has a base associated with it. The most common bases used are 10 (log base 10) and the natural logarithm with base e (denoted as ln). The base determines the behavior and properties of the logarithm function.The general form of the logarithm function is expressed as: logb(x)Here, "b" represents the base, and "x" is the argument of the logarithm. The base determines the scaling factor by which the argument is raised to reach a given value. In other words, it tells us what power we need to raise the base to obtain the argument.The logarithm function essentially answers the question, "To what power must the base be raised to obtain the argument?" For example, in base 10 logarithm, log10(100) = 2, because 10 raised to the power of 2 equals 100.The logarithm function has several important properties, such as the logarithmic identities and rules, which allow for simplification and manipulation of logarithmic expressions. It is widely used in various fields, including mathematics, physics, engineering, computer science, and finance, among others.