Log Calculator
Free Log Calculator for algebra. Enter values to get step-by-step solutions with formulas and graphs. See charts, tables, and visual results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
log_b(x) = ln(x) / ln(b)
Where b is the base of the logarithm, x is the value, and ln is the natural logarithm. This change-of-base formula allows computing a logarithm of any base using the natural logarithm function.
Worked Examples
Example 1: Common Logarithm Calculation
Problem:Find log base 10 of 5000.
Solution:Using the change of base formula or direct computation:\nlog10(5000) = log10(5 x 1000) = log10(5) + log10(1000)\nlog10(5) = 0.69897\nlog10(1000) = 3\nlog10(5000) = 0.69897 + 3 = 3.69897
Result:log10(5000) = 3.69897, meaning 10 raised to 3.69897 equals 5000
Example 2: Solving an Exponential Equation
Problem:How many years does it take for an investment to triple at 8% annual growth? Solve 1.08^t = 3.
Solution:Take the natural log of both sides:\nln(1.08^t) = ln(3)\nt x ln(1.08) = ln(3)\nt = ln(3) / ln(1.08)\nt = 1.09861 / 0.07696\nt = 14.275 years
Result:t = 14.275 years for the investment to triple at 8% annual growth
Frequently Asked Questions
What is the difference between natural log, common log, and binary log?
The three most commonly used logarithm bases each serve different purposes. The common logarithm (log base 10) is used extensively in science and engineering for measuring things like pH, decibels, and earthquake magnitude on the Richter scale. The natural logarithm (ln, base e where e is approximately 2.71828) appears naturally in calculus, continuous growth models, and physics equations. The binary logarithm (log base 2) is essential in computer science for analyzing algorithm complexity, data structures, and information theory. Each base is simply a scaled version of the others via the change-of-base formula.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy