Enter Calculator Values
Choose a mode, enter values, then calculate.
Keyboard shortcut: press Ctrl + Enter to calculate.
Calculation History and Favorites
History stays inside this browser.
Formula Used
These identities power the calculator modes.
Common logarithm
y = log₁₀(x) ⇔ 10ʸ = xThe real-number domain requires x greater than zero.
Antilogarithm
x = 10ʸThis operation reverses the common logarithm.
Change of base
log_b(x) = log₁₀(x) / log₁₀(b)The base stays positive and cannot equal one.
Product rule
log₁₀(ab) = log₁₀(a) + log₁₀(b)Both factors must remain positive.
Quotient rule
log₁₀(a/b) = log₁₀(a) − log₁₀(b)Both quotient values must remain positive.
Power rule
log₁₀(aⁿ) = n log₁₀(a)The exponent moves before the logarithm.
pH scale
pH = −log₁₀[H⁺]Concentration uses moles per liter.
Decibel power ratio
L = 10 log₁₀(P₁/P₀)Power ratios use a multiplier of ten.
Decibel amplitude ratio
L = 20 log₁₀(A₁/A₀)Amplitude ratios commonly use twenty.
Complex principal value
log₁₀(z) = ln|z|/ln(10) + i·arg(z)/ln(10)The principal argument lies inside its standard interval.
How to Use This Calculator
Follow these steps for reliable results.
Select a calculation mode
Choose a common logarithm, equation, rule, or application.
Enter the requested values
Use decimals, scientific notation, or supported arithmetic expressions.
Choose result precision
Select zero through fifteen decimal places.
Calculate and inspect
Review the answer, details, steps, and graph.
Export or save
Copy, print, download, share, or favorite results.
Example Data Table
These values show common logarithm patterns.
| Input x | log₁₀(x) | Exponential form | Characteristic | Mantissa |
|---|---|---|---|---|
| 0.0001 | −4 | 10⁻⁴ = 0.0001 | −4 | 0 |
| 0.001 | −3 | 10⁻³ = 0.001 | −3 | 0 |
| 0.01 | −2 | 10⁻² = 0.01 | −2 | 0 |
| 0.1 | −1 | 10⁻¹ = 0.1 | −1 | 0 |
| 1 | 0 | 10⁰ = 1 | 0 | 0 |
| 2 | 0.30103 | 10⁰·³⁰¹⁰³ ≈ 2 | 0 | 0.30103 |
| 5 | 0.69897 | 10⁰·⁶⁹⁸⁹⁷ ≈ 5 | 0 | 0.69897 |
| 10 | 1 | 10¹ = 10 | 1 | 0 |
| 100 | 2 | 10² = 100 | 2 | 0 |
| 250 | 2.39794 | 10²·³⁹⁷⁹⁴ ≈ 250 | 2 | 0.39794 |
| 1,000 | 3 | 10³ = 1,000 | 3 | 0 |
| 1,000,000 | 6 | 10⁶ = 1,000,000 | 6 | 0 |
Understanding Base-10 Logarithms
A base-ten logarithm finds a decimal exponent. It asks which power creates a positive number. For example, ten cubed equals one thousand. Therefore, log₁₀(1000) equals three. This direct relationship explains every common logarithm.
Domain and range
Real common logarithms accept only positive arguments. Zero has no finite logarithm. Negative inputs require complex-number methods. The real output may be any number. Large inputs produce positive logarithms. Values between zero and one produce negative logarithms.
Characteristic and mantissa
The characteristic is the logarithm's integer floor. The mantissa is its remaining fractional part. This convention keeps mantissas nonnegative. For 250, the logarithm is about 2.39794. Its characteristic is two. Its mantissa is approximately 0.39794.
Powers and scientific notation
Powers of ten create especially simple answers. Each decimal shift changes the logarithm by one. Scientific notation exposes this pattern clearly. For 5 × 10⁸, apply the product rule. The answer becomes log₁₀(5) plus eight.
Logarithm laws
Logarithm laws simplify multiplication, division, and powers. A product becomes a sum. A quotient becomes a difference. An exponent becomes a multiplier. These transformations help solve equations efficiently. They also reduce difficult numeric calculations.
Inverse calculations
The antilogarithm reverses a common logarithm. Raise ten to the logarithm result. This operation returns the original positive input. It also verifies rounded answers. Small differences can appear after rounding.
Graphs and behavior
The logarithm graph passes through one, zero. It also passes through ten, one. The curve increases slowly forever. It never crosses the vertical axis. That axis forms a vertical asymptote. The inverse exponential graph rises much faster.
Practical scales
Common logarithms compress wide numeric ranges. Decibel formulas compare signal power or amplitude. Chemistry uses logarithms for pH. Magnitude scales compare huge intensity differences. Orders of magnitude summarize scale changes quickly.
Equation solving
Convert logarithmic equations into exponential form first. Then isolate the unknown using ordinary algebra. Always check the original domain afterward. Extraneous values can violate positive-argument requirements. The calculator performs this validation automatically.
Accurate result handling
Keep extra digits during intermediate calculations. Round only the final result. Scientific notation helps prevent visual mistakes. Choose precision matching the original measurements. More displayed digits do not guarantee greater accuracy.
Common mistakes
Common mistakes begin with ignored domain restrictions. Never enter zero in real-number mode. Avoid confusing log₁₀ with natural logarithms. Check whether software labels log consistently. Parentheses matter during equation entry. Write numerator and denominator groups. Keep units consistent before using application modes. Decibel power and amplitude formulas use different multipliers. Using ten instead of twenty changes conclusions substantially. Do not round intermediate logarithms too early. Early rounding can distort inverse checks. Record enough digits for later calculations. Finally, confirm every solution in the original equation. Domain testing removes values created during algebraic manipulation. Consistent notation makes logarithmic work easier.
Frequently Asked Questions
Review common questions about base-ten logarithms.
What is a base-10 logarithm?
It is the exponent applied to ten. That exponent produces the entered positive value.
Are log and log₁₀ always identical?
Many elementary contexts use log for base ten. Some technical contexts use another convention.
Why is log₁₀(0) undefined?
No finite power of ten equals zero. The graph approaches zero without reaching it.
Can I calculate a negative input?
Negative inputs have no real common logarithm. Use the provided complex-number mode instead.
Why do small decimals have negative logarithms?
Negative powers of ten create positive values below one.
What is an antilogarithm?
An antilogarithm raises ten to a chosen exponent. It reverses log₁₀.
What is the change-of-base formula?
Divide log₁₀ of the argument by log₁₀ of the target base.
What are characteristic and mantissa?
The characteristic is the integer floor. The mantissa is the nonnegative fractional remainder.
How does scientific notation simplify logarithms?
It separates a coefficient from a power of ten. Their logarithms then add.
How many decimal places should I use?
Match precision with your source data. Keep extra digits before final rounding.
Can I save or export calculations?
Yes. Copy results, export CSV files, print reports, and store browser favorites.
Privacy and Calculation Notes
Important behavior is explained below.
Server calculation
Submitted values are processed during the current page request.
Browser history
History and favorites use local browser storage.
Numeric limits
Extremely large powers may exceed server numeric limits.
Educational purpose
Application results require proper scientific context.