Log Base 10 Calculator
Calculate common logarithms with step-by-step solutions and antilog conversions
Expression:
log₁₀() 10 =
log₁₀() 10
Step-by-Step Solution
Natural Log (ln)
ln()
Log Base 2
log₂()
Antilog Result
10^
Logarithm Properties Reference
| Property | Formula |
|---|---|
Common Log Values
Real-World Applications
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About Log Base 10 Calculator
What is Log Base 10?
The common logarithm (log base 10) is the power to which 10 must be raised to obtain a given number. If log₁₀(x) = y, then 10^y = x.
How to Use This Calculator
- Select your calculation mode: Choose from Calculate Log₁₀, Antilog, or Solve Equation
- Enter your value: Input the number you want to calculate
- View instant results: See the result with step-by-step explanation
- Explore properties: Learn logarithm rules and applications
Key Formulas
Logarithm Definition
- If 10^y = x, then log₁₀(x) = y
- Example: log₁₀(1000) = 3 because 10³ = 1000
Antilogarithm (Inverse)
- If log₁₀(x) = y, then x = 10^y
- Example: antilog(2) = 10² = 100
Logarithm Properties
Product Rule
log₁₀(xy) = log₁₀(x) + log₁₀(y)
Quotient Rule
log₁₀(x/y) = log₁₀(x) - log₁₀(y)
Power Rule
log₁₀(x^n) = n × log₁₀(x)
Change of Base
log_b(x) = log₁₀(x) / log₁₀(b)
Real-World Applications
Decibel Scale (Sound)
Sound intensity is measured in decibels: dB = 10 × log₁₀(I/I₀)
Richter Scale (Earthquakes)
Earthquake magnitude uses logarithms: each whole number increase represents 10× more amplitude.
pH Scale (Chemistry)
pH = -log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration.
Scientific Notation
log₁₀ helps determine the order of magnitude of numbers.
Common Values
| x | log₁₀(x) |
|---|---|
| 0.001 | -3 |
| 0.01 | -2 |
| 0.1 | -1 |
| 1 | 0 |
| 10 | 1 |
| 100 | 2 |
| 1000 | 3 |
Note: Logarithms are only defined for positive numbers. log₁₀(0) and log₁₀(negative) are undefined in real numbers.