Common Logarithms (Base 10)

Learn the common logarithm log x with base 10: exact values for powers of 10, estimating logs between whole numbers, and using a calculator.

Advanced25 min18 problems
Practice Problems

Definition

The common logarithm is the logarithm with base 10. It is so common that we leave the base out:
log⁡x=log⁡10x\log x = \log_{10} x
log⁡x\log x answers the question: **"10 to which power gives xx?"**
log⁡x=n  ⟺  10n=x\log x = n \iff 10^n = x
Powers of 10 give whole-number logs:
xx0.0010.0010.010.010.10.11110101001001 0001\,0001 000 0001\,000\,000
log⁡x\log x−3-3−2-2−1-10011223366
So log⁡(10n)=n\log(10^n) = n for every exponent nn.
Numbers between two powers of 10 have a log between two whole numbers. Since 10<50<10010 < 50 < 100, we know 1<log⁡50<21 < \log 50 < 2. A calculator (the LOG key) gives log⁡50≈1.699\log 50 \approx 1.699.
The argument must be positive: log⁡0\log 0 and log⁡(−5)\log(-5) do not exist.

Try it now

What is log⁡10 000\log 10\,000?

Worked Examples

Find log⁡100 000\log 100\,000 and log⁡1\log 1 without a calculator.
1

Ask the question for 100 000

10 to which power gives 100 000100\,000? → 10n=100 00010^n = 100\,000
2

Count the zeros

100 000100\,000 is a 1 followed by 5 zeros, so 100 000=105100\,000 = 10^5 → log⁡100 000=5\log 100\,000 = 5
3

Ask the question for 1

Any nonzero number to the power 0 is 1: 100=110^0 = 1 → log⁡1=0\log 1 = 0

Common Mistakes

Thinking log⁡50=1.5\log 50 = 1.5 because 50 is about halfway between 10 and 100

Why it's wrong: A log scale is not evenly spaced. Each step of 1 in the log multiplies the number by 10, so the halfway point between 10 and 100 is 101.5≈31.610^{1.5} \approx 31.6.

Correct: log⁡50≈1.699\log 50 \approx 1.699. Estimate the two whole numbers first (1<log⁡50<21 < \log 50 < 2), then use the calculator.

Writing log⁡0.01=2\log 0.01 = 2 or saying the log of a decimal does not exist

Why it's wrong: 0.01=10−20.01 = 10^{-2}, so the exponent is negative. Logs of numbers between 0 and 1 exist, they are just negative.

Correct: log⁡0.01=−2\log 0.01 = -2, log⁡0.1=−1\log 0.1 = -1, log⁡0.001=−3\log 0.001 = -3.

Writing log⁡0=0\log 0 = 0 or log⁡(−100)=−2\log(-100) = -2

Why it's wrong: 10n10^n is positive for every exponent nn. No power of 10 equals 0 or a negative number.

Correct: log⁡0\log 0 and log⁡(−100)\log(-100) are undefined. The calculator shows an error. Remember: log⁡1=0\log 1 = 0, not log⁡0\log 0.

Counting all the digits: log⁡1 000=4\log 1\,000 = 4

Why it's wrong: 1 0001\,000 has 4 digits but only 3 zeros. The log counts the factors of 10, not the digits.

Correct: log⁡1 000=3\log 1\,000 = 3 because 103=1 00010^3 = 1\,000. A whole number with dd digits has a log between d−1d - 1 and dd.

Interactive Visual

10³ = 1000
3

See how powers of a number grow on the number line. Change the base and exponent.

Bar Chart

Common logarithm of x

0.01(-2)
0.1(-1)
1(0)
5(0.7)
10(1)
50(1.7)
100(2)
1000(3)

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is log⁡10 000\log 10\,000?

Why It Matters

Base 10 is the base of our number system, so the common logarithm tells you the size of a number at a glance:
  • Orders of magnitude: the whole-number part of log⁡x\log x is the exponent of xx in scientific notation. log⁡(3 000)≈3.477\log(3\,000) \approx 3.477 and 3 000=3×1033\,000 = 3 \times 10^3.
  • Chemistry: the pH of a solution is −log⁡-\log of its hydrogen ion concentration.
  • Sound: decibels use 10log⁡10 \log of an intensity ratio.
  • Earthquakes and stars: the Richter scale and star brightness scales are built on base 10 logs.
  • Science graphs: log scales put 11, 1010, 100100 and 1 000 0001\,000\,000 on one readable axis.
Knowing the logs of powers of 10 by heart lets you check any calculator answer in seconds.

Real World Applications

Orders of Magnitude

Scientists compare huge and tiny quantities by their order of magnitude: the whole-number part of their common logarithm.

Example:

A human hair is about 0.00010.0001 m thick, and log⁡0.0001=−4\log 0.0001 = -4. The Earth is about 10 000 00010\,000\,000 m across, and log⁡10 000 000=7\log 10\,000\,000 = 7. The difference of 11 orders of magnitude means the Earth is about 101110^{11} times wider.

1Try It Yourself

The world population is about 8 000 000 0008\,000\,000\,000 people.

What is the order of magnitude, that is the whole-number part of log⁡8 000 000 000\log 8\,000\,000\,000?

Step 1: Write the mathematical expression

Write the whole-number part of the log:

pH in Chemistry

The pH of a solution is $\text{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration in mol/L. Lower pH means more acidic.

Example:

Pure water has [H+]=0.0000001=10−7[H^+] = 0.0000001 = 10^{-7}, so pH=−(−7)=7\text{pH} = -(-7) = 7 (neutral).

2Try It Yourself

A sample of orange juice has [H+]=0.0001[H^+] = 0.0001 mol/L.

What is its pH?

Step 1: Write the mathematical expression

Calculate −log⁡0.0001-\log 0.0001:

Decibels and Sound

Sound level in decibels is $L = 10 \log\left(\dfrac{I}{I_0}\right)$, where $\dfrac{I}{I_0}$ says how many times stronger a sound is than the quietest sound we can hear.

Example:

A whisper is about 100100 times the threshold: L=10log⁡100=10×2=20L = 10 \log 100 = 10 \times 2 = 20 dB.

3Try It Yourself

Normal conversation is about 1 000 0001\,000\,000 times as intense as the threshold of hearing.

What is its sound level in decibels?

Step 1: Write the mathematical expression

Calculate 10log⁡1 000 00010 \log 1\,000\,000:

Key Takeaways

  • 1The common logarithm has base 10: log⁡x=log⁡10x\log x = \log_{10} x, and log⁡x=n\log x = n means 10n=x10^n = x
  • 2Powers of 10 have whole-number logs: log⁡(10n)=n\log(10^n) = n, for example log⁡1 000=3\log 1\,000 = 3 and log⁡0.01=−2\log 0.01 = -2
  • 3log⁡1=0\log 1 = 0 and log⁡10=1\log 10 = 1; numbers between 0 and 1 have negative logs
  • 4To estimate log⁡x\log x, find the powers of 10 around xx: 10<50<10010 < 50 < 100 gives 1<log⁡50<21 < \log 50 < 2
  • 5The whole-number part of log⁡x\log x is the order of magnitude, the exponent of xx in scientific notation
  • 6Use the LOG key for exact decimals (log⁡50≈1.699\log 50 \approx 1.699); log⁡0\log 0 and logs of negative numbers do not exist

Frequently Asked Questions

Our number system is built on 10, so base 10 logs were the ones printed in log tables for centuries and the ones on every scientific calculator (the LOG key). Because they are used so often, the base is simply left out: log⁡x\log x means log⁡10x\log_{10} x.
Our number system is built on 10, so base 10 logs were the ones printed in log tables for centuries and the ones on every scientific calculator (the LOG key). Because they are used so often, the base is simply left out: log⁡x\log x means log⁡10x\log_{10} x.
It uses a fast numeric method to find the exponent nn with 10n=5010^n = 50. You can check its answer: 101.699≈5010^{1.699} \approx 50. The estimate 1<log⁡50<21 < \log 50 < 2 tells you if you pressed the right keys.
Numbers between 0 and 1 are reached with negative exponents: 0.01=1100=10−20.01 = \dfrac{1}{100} = 10^{-2}. So log⁡0.01=−2\log 0.01 = -2. Only the argument must be positive, the log itself can be any real number.
No. LOG uses base 10, while LN uses a different base, the number e≈2.718e \approx 2.718. For example log⁡100=2\log 100 = 2, but ln⁡100≈4.605\ln 100 \approx 4.605. The lesson Natural Logarithms (ln) covers that key.

Glossary

Common logarithm
The logarithm with base 10, written log⁡x\log x: the exponent nn with 10n=x10^n = x
Base
The number that is raised to a power; for the common logarithm it is 10
Argument
The number you take the logarithm of, the xx in log⁡x\log x; it must be positive
Power of 10
A number of the form 10n10^n, such as 1 000=1031\,000 = 10^3 or 0.01=10−20.01 = 10^{-2}
Order of magnitude
The whole-number part of log⁡x\log x, which is the exponent of xx in scientific notation
Scientific notation
Writing a number as a×10na \times 10^n with 1≤a<101 \le a < 10, such as 3 000=3×1033\,000 = 3 \times 10^3

Formula Card

Common logarithm

log⁡x=n  ⟺  10n=x\log x = n \iff 10^n = x

The base 10 is not written

Log of a power of 10

log⁡(10n)=n\log(10^n) = n

For example $\log 1\,000 = 3$, $\log 0.01 = -2$

Special values

log⁡1=0,log⁡10=1\log 1 = 0, \quad \log 10 = 1

Because $10^0 = 1$ and $10^1 = 10$

Estimating

10n≤x<10n+1⇒n≤log⁡x<n+110^n \le x < 10^{n+1} \Rightarrow n \le \log x < n + 1

The powers of 10 around $x$ give the whole-number part

Back to the number

10log⁡x=x10^{\log x} = x

For example $10^{1.699} \approx 50$

Today's challengeOne problem, three tries

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Introduction to Logarithms

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Natural Logarithms (ln)

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Graphing Logarithmic Functions

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