Mathwords logoMathwords

Common Logarithm — Definition, Formula & Examples

Common Logarithm

The logarithm base 10 of a number. That is, the power of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 102 = 100.

 

 

See also

Natural logarithm, logarithm rules

Key Formula

log⁡x=y⟺10y=x\log x = y \quad \Longleftrightarrow \quad 10^y = x
Where:
  • xx = The positive number you are taking the logarithm of (called the argument)
  • yy = The exponent that 10 must be raised to in order to equal x

Worked Example

Problem: Find the common logarithm of 5000.
Step 1: Rewrite the problem using the definition. You need to find y such that 10^y = 5000.
log⁡5000=y⟺10y=5000\log 5000 = y \quad \Longleftrightarrow \quad 10^y = 5000
Step 2: Express 5000 as a product of simpler factors whose logarithms you know.
5000=5×1000=5×1035000 = 5 \times 1000 = 5 \times 10^3
Step 3: Apply the logarithm product rule: log(ab) = log a + log b.
log⁡5000=log⁡5+log⁡103=log⁡5+3\log 5000 = \log 5 + \log 10^3 = \log 5 + 3
Step 4: Use the known value log 5 ≈ 0.6990 (from a calculator or table).
log⁡5000≈0.6990+3=3.6990\log 5000 \approx 0.6990 + 3 = 3.6990
Step 5: Verify: raise 10 to this power to check.
103.6990≈5000  ✓10^{3.6990} \approx 5000 \; \checkmark
Answer: log 5000 ≈ 3.699

Another Example

Problem: Solve for x: log x = −2.
Step 1: Convert the logarithmic equation to exponential form using the definition.
log⁡x=−2⟺10−2=x\log x = -2 \quad \Longleftrightarrow \quad 10^{-2} = x
Step 2: Evaluate the power of 10.
x=10−2=1102=1100=0.01x = 10^{-2} = \frac{1}{10^2} = \frac{1}{100} = 0.01
Answer: x = 0.01

Frequently Asked Questions

What is the difference between log and ln?
"log" (without a base) refers to the common logarithm, which uses base 10. "ln" refers to the natural logarithm, which uses base e ≈ 2.718. They answer the same type of question—'what exponent gives this number?'—but with different bases. You can convert between them using ln x = (log x) / (log e) ≈ (log x) / 0.4343.
Why is log 0 undefined?
Because there is no power of 10 that equals 0. No matter how large and negative you make the exponent, 10 raised to that power gets closer and closer to 0 but never reaches it. For the same reason, you cannot take the common logarithm of any negative number.

Common Logarithm (log) vs. Natural Logarithm (ln)

Common Logarithm (log)Natural Logarithm (ln)
BaseBase 10Base e ≈ 2.71828
Notationlog x or log₁₀ xln x or logₑ x
Typical useScales of measurement (pH, decibels, Richter), general computationCalculus, continuous growth/decay, pure mathematics
Key valuelog 10 = 1ln e = 1

Why It Matters

The common logarithm is the foundation of many real-world scales that compress huge ranges of values into manageable numbers. The Richter scale for earthquakes, the decibel scale for sound intensity, and pH for acidity all rely on base-10 logarithms. Understanding common logarithms also prepares you for scientific notation and for solving exponential equations that appear throughout science and finance.

Common Mistakes

Mistake: Thinking that log(a + b) equals log a + log b.
Correction: The product rule says log(a · b) = log a + log b. There is no simple rule for the logarithm of a sum. For example, log(100 + 1000) = log 1100 ≈ 3.041, which is not log 100 + log 1000 = 2 + 3 = 5.
Mistake: Forgetting that the argument of a logarithm must be positive.
Correction: You can only take log x when x > 0. Expressions like log(0) and log(−5) are undefined in the real numbers because no real power of 10 produces zero or a negative result.

Related Terms