Log Calculator (Logarithm)
Calculate the logarithm of a number in any base. Computes log_b(x) — the power to which base b must be raised to produce x.
This free online log calculator (logarithm) provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.
Must be positive
Must be positive and ≠ 1
Results
log_b(x)
2
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Log Calculator (Logarithm). Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.
Review your inputs
Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Log Calculator (Logarithm) instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
Formula Reference
Log Calculator (Logarithm) Formula
See calculator inputs for the governing equation
Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.
When to Use This Calculator
- •Use the Log Calculator (Logarithm) when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
About This Calculator
The Log Calculator (Logarithm) is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the logarithm of a number in any base. Computes log_b(x) — the power to which base b must be raised to produce x. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.
About Log Calculator (Logarithm)
The logarithm is the inverse of exponentiation. If bʸ = x, then log_b(x) = y. Logarithms answer the question: to what power must I raise the base b to get x? They were invented by John Napier in the early 17th century as a computational tool for simplifying multiplication into addition. Today, logarithms are indispensable in science (pH scale, Richter scale, decibels), computer science (algorithm complexity, information theory), finance (continuous compounding), and mathematics (calculus, number theory). The three most common bases are 10 (common logarithm), e ≈ 2.718 (natural logarithm), and 2 (used in computer science). This calculator computes logarithms in any positive base using the change-of-base formula.
The Math Behind It
Formula Reference
Change of Base
log_b(x) = ln(x) / ln(b)
Variables: b = base, x = argument
Definition
If b^y = x, then log_b(x) = y
Variables: b > 0, b ≠ 1, x > 0
Worked Examples
Example 1: Calculate log₁₀(1000)
Find the base-10 logarithm of 1000.
log₁₀(1000) = 3
Example 2: Calculate log₂(64)
Find the base-2 logarithm of 64.
log₂(64) = 6
Common Mistakes & Tips
- !Trying to take the logarithm of zero or a negative number — it is undefined for real numbers.
- !Confusing log (base 10), ln (base e), and lg (base 2).
- !Misapplying the product rule: log(x + y) ≠ log(x) + log(y); the rule is log(xy) = log(x) + log(y).
- !Forgetting that log_b(1) = 0 for all bases b.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
What is the difference between log, ln, and lg?
log typically means base 10 (common log), ln means base e (natural log), and lg means base 2. However, in pure mathematics, 'log' often refers to the natural logarithm.
Why can't you take the log of a negative number?
In the real number system, no real power of a positive base can produce a negative result, so the logarithm is undefined. In complex analysis, logarithms of negative numbers exist but are complex-valued.