What is a Logarithm?
A logarithm is a mathematical function that determines the power to which a base must be
raised to produce a given number. Logarithms are the inverse operation of exponentiation.
For example, if \( 10^2 = 100 \), then the logarithm base 10 of 100 is 2, written as \(
\log_{10}(100) = 2 \).
Logarithms are widely used in science, engineering, and mathematics to simplify complex
calculations, such as those involving exponential growth, sound intensity (decibels), pH
levels in chemistry, and even in algorithms for computer science.
Our logarithm calculator makes this process easy by allowing you to compute logarithms with
different bases (base 10, base \( e \), or a custom base) and even logarithms of
trigonometric functions like \( \sin \), \( \cos \), and \( \tan \).
Formulas for Calculating Logarithms
Standard Logarithms
The logarithm of a number \( x \) with base \( b \) is given by:
\( \log_b(x) = y \quad \text{such that} \quad b^y = x \)
Common logarithms use base 10:
\( \log_{10}(x) \)
Natural logarithms use base \( e \) (approximately 2.71828):
\( \ln(x) = \log_e(x) \)
For a custom base, the change of base formula is used:
\( \log_b(x) = \frac{\ln(x)}{\ln(b)} \)
Logarithms of Trigonometric Functions
To calculate the logarithm of a trigonometric function, first compute the trigonometric
value, then apply the logarithm. For example, to find \( \log_{10}(\sin(\theta)) \):
- Compute \( \sin(\theta) \).
- Ensure \( \sin(\theta) > 0 \) (since logarithms are undefined for non-positive numbers).
- Calculate \( \log_{10}(\sin(\theta)) \).
Steps to Calculate a Logarithm
For a Direct Number
1
Choose the Base: Select the base of the logarithm (e.g., 10, \( e \),
or a custom base).
2
Input the Number: Enter the number \( x \) (must be positive).
3
Compute the Logarithm: Use the appropriate formula based on the base.
For a Trigonometric Function
1
Select the Function and Angle: Choose the trigonometric function (\(
\sin \), \( \cos \), or \( \tan \)) and the angle (in degrees or radians).
2
Compute the Trigonometric Value: Calculate the value of the function
(e.g., \( \sin(\theta) \)).
3
Check Validity: Ensure the result is positive and finite.
4
Apply the Logarithm: Compute the logarithm of the trigonometric result
using the chosen base.
Examples: Calculating Logarithms
Example 1: Direct Number
Example: Calculate \( \log_{10}(100) \).
- Step 1: Base is 10, number is 100.
- Step 2: Compute the logarithm: \( \log_{10}(100) \).
- Since \( 10^2 = 100 \), we have \( \log_{10}(100) = 2 \).
So, \( \log_{10}(100) = 2 \).
Example 2: Trigonometric Function
Example: Calculate \( \ln(\sin(30^\circ)) \).
- Step 1: Compute \( \sin(30^\circ) \). Since \( \sin(30^\circ) = 0.5 \).
- Step 2: Check validity: 0.5 is positive and finite.
- Step 3: Calculate the natural logarithm: \( \ln(0.5) \).
- Using the value of \( \ln(0.5) \approx -0.693147 \).
So, \( \ln(\sin(30^\circ)) \approx -0.693147 \).
Features of Our Logarithm Calculator
Our logarithm calculator, as of May 16, 2025, offers the following features:
- Multiple Bases: Compute common logarithms (base 10), natural logarithms
(base \( e \)), or logarithms with a custom base.
- Trigonometric Support: Calculate logarithms of trigonometric functions
(\( \sin \), \( \cos \), \( \tan \)) with angles in degrees or radians.
- Error Handling: Validates inputs to ensure numbers are positive and
trigonometric results are valid for logarithms (e.g., handles cases like \(
\tan(90^\circ) \) by returning "Infinity").
- Responsive Design: Built with Bootstrap for a clean, mobile-friendly
interface.
Whether you're a student, engineer, or enthusiast, this calculator simplifies logarithm
calculations for both standard numbers and trigonometric functions.
