Logarithmic Calculator

Input the base and the number, and quickly calculate the logarithmic value.

Calculate Logarithm: logbN

Result

What Is a Logarithm?

A logarithm is a mathematical tool that solves exponential equations. It helps determine how many times a base number must be multiplied by itself to reach another number. For example, given a base \( b \) and a number \( N \), the logarithmic formula is: \( \log_b(N) = x \) This means \( b^x = N \). Logarithms are widely used in fields such as scientific computation and data analysis.

Common Types of Logarithms

  • Binary logarithms: Logarithms with base 2, denoted as \( \text{lb} \).
  • Natural logarithms: Logarithms with base \( e \) (Euler's number), denoted as \( \ln \).
  • Common logarithms: Logarithms with base 10, also called standard logarithms, denoted as \( \lg \).

Examples

Example 1: Calculate the logarithm of 100 with base 10

Solution:

\( \log_{10}(100) = x \)

Since \( 10^2 = 100 \), \( x = 2 \).

Result: The logarithm of 100 with base 10 is 2.

Example 2: Calculate the logarithm of 625 with base 5

Solution:

\( \log_5(625) = x \)

Since \( 5^4 = 625 \), \( x = 4 \).

Result: The logarithm of 625 with base 5 is 4.

Example 3: Calculate the natural logarithm of \( e^3 \)

Solution:

\( \ln(e^3) = 3 \)

Result: The natural logarithm of \( e^3 \) is 3.