Common Logarithm- Properties and How to Use
What Are Common Logarithms?
Common logarithms are logarithms with base 10. They're written as log(x) instead of log₁₀(x). This is the logarithm most people encounter in high school math and real-world applications.
When you see log(100), it's asking: "10 raised to what power gives 100?" The answer is 2. Simple enough.
The "common" part refers to base 10 because that's the number system humans use. Natural logarithms use base e (approximately 2.718), but that's a different beast.
The Core Definition
If y = 10ˣ, then log(y) = x.
This relationship works both ways. You can swap between exponential and logarithmic forms, and the equation stays true.
Quick Reference
- 10⁰ = 1, so log(1) = 0
- 10¹ = 10, so log(10) = 1
- 10² = 100, so log(100) = 2
- 10³ = 1000, so log(1000) = 3
- 10⁻¹ = 0.1, so log(0.1) = -1
Notice the pattern. Each step up multiplies by 10. Each step down divides by 10. The logarithm of any number between 1 and 10 falls somewhere between 0 and 1.
Key Properties of Common Logarithms
These properties let you break down complex log problems into simpler pieces. Memorize them. You'll use them constantly.
Product Property
log(MN) = log(M) + log(N)
The log of a product equals the sum of the logs. This works because you're adding exponents when multiplying numbers with the same base.
Example: log(6) = log(2 × 3) = log(2) + log(3)
Quotient Property
log(M/N) = log(M) - log(N)
The log of a quotient equals the difference of the logs. Subtracting exponents when dividing.
Example: log(5) = log(10/2) = log(10) - log(2) = 1 - log(2)
Power Property
log(Mᵖ) = p × log(M)
When raising a number to a power, you can bring that exponent down and multiply. This is the most useful property for solving equations.
Example: log(8) = log(2³) = 3 × log(2)
Change of Base (When You Need a Different Base)
Sometimes you need to calculate logs with bases other than 10. Use this formula:
logₐ(x) = log(x) / log(a)
You can use either natural log or common log on the right side. The ratio stays the same.
How to Use Common Logarithms: Practical Examples
Solving Simple Equations
Example 1: Find x if log(x) = 3
This asks: "10 to what power equals x?" Answer: x = 10³ = 1000
Example 2: Find x if log(x) = -2
x = 10⁻² = 0.01
Solving Exponential Equations
Example: Solve 10ˣ = 500
Take log of both sides:
log(10ˣ) = log(500)
Using the power property: x × log(10) = log(500)
Since log(10) = 1: x = log(500)
Using a calculator: x ≈ 2.699
You can verify: 10²·⁶⁹⁹ ≈ 500 ✓
Expanding Logarithmic Expressions
Example: Expand log(50)
Write 50 as 5 × 10:
log(50) = log(5 × 10) = log(5) + log(10) = log(5) + 1
Example: Expand log(0.04)
Write 0.04 as 4/100 or 4 × 10⁻²:
log(0.04) = log(4) + log(10⁻²) = log(4) - 2
Condensing Logarithmic Expressions
Example: Condense log(3) + log(7) - log(2)
Combine the addition: log(3 × 7) = log(21)
Then the subtraction: log(21) - log(2) = log(21/2)
Final answer: log(10.5)
Logarithm Properties Comparison
| Property | Formula | Example |
|---|---|---|
| Product | log(MN) = log(M) + log(N) | log(6) = log(2) + log(3) |
| Quotient | log(M/N) = log(M) - log(N) | log(5) = log(10) - log(2) |
| Power | log(Mᵖ) = p × log(M) | log(8) = 3 × log(2) |
| Zero | log(1) = 0 | 10⁰ = 1 |
| Identity | log(10) = 1 | 10¹ = 10 |
Getting Started: Solving Your First Log Problem
Follow these steps when you encounter a logarithm problem:
- Identify the base. If it's just "log" with no subscript, assume base 10.
- Convert to exponential form if solving for a variable. Remember: log₁₀(x) = y means 10ʸ = x.
- Apply properties to break down complex expressions before calculating.
- Use your calculator for messy numbers. Make sure you know where the log button is.
Practice problem: Solve for x: 10ˣ = 250
Take log of both sides: x = log(250)
Calculate: x ≈ 2.398
Common Mistakes to Avoid
- Confusing log and ln. log is base 10. ln is base e. Different numbers, different answers.
- Forgetting the power property. log(x²) is NOT 2×log(x). Wait—actually it is. That's the whole point. Use it.
- Taking log of a negative number. Doesn't work in real numbers. If your equation gives you log(-3), something went wrong.
- Dropping parentheses. log(x + y) is not the same as log(x) + y. The log applies to the entire expression inside.
Where Common Logarithms Actually Appear
These aren't just textbook exercises. Common logs show up in real situations:
- Decibel scale for sound intensity uses log₁₀
- pH in chemistry is -log₁₀ of hydrogen ion concentration
- Richter scale for earthquake magnitude uses log₁₀
- Acid dissociation constants often expressed as pKa = -log₁₀(Ka)
When scientists say "log scale," they mean base 10. It's everywhere once you start looking.