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.
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.
How accurate are the results from Log Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
How do I get the most accurate result?
Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.
Is Log Calculator free to use?
Yes, completely free with no sign-up required. All calculators on NovaCalculator are free to use without registration, subscription, or payment.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.