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Teacher Guide: 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.

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Class quiz

10 questions on Logarithms. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Read log⁡x\log x as the logarithm with base 10
  • Evaluate log⁡(10n)=n\log(10^n) = n for powers of 10 from 0.0010.001 to 1 000 0001\,000\,000 without a calculator
  • Estimate the log of any positive number between two whole numbers
  • Use the LOG key of a calculator and check the result with an estimate
  • Connect the whole-number part of log⁡x\log x to scientific notation and orders of magnitude
Prerequisites
  • • Introduction to Logarithms and Definition of a Logarithm: log⁡by=x  ⟺  bx=y\log_b y = x \iff b^x = y
  • • Powers of 10, including negative exponents (10−2=0.0110^{-2} = 0.01)
  • • Scientific notation (3 000=3×1033\,000 = 3 \times 10^3)
Discussion Starters
  • 1. Why do you think base 10 logs were the first to be printed in tables and put on calculators?
  • 2. If log⁡x\log x goes up by 1, what happens to xx? And if it goes up by 2?
  • 3. 5050 is about halfway between 1010 and 100100. Why is log⁡50\log 50 not 1.51.5?
  • 4. How could you guess the number of digits of a big whole number from its log?
  • 5. Where in science or daily life have you seen a scale where each step means 10 times more?
Common Misconceptions

The log of a number between 10 and 100 grows evenly, so log⁡50=1.5\log 50 = 1.5

Remediation: Show the log scale number line: equal steps mean multiplying by 10. Compute 101.5≈31.610^{1.5} \approx 31.6 on the calculator, so the middle of the scale is 31.6, not 55.

Logs of decimals are undefined, or log⁡0.01=2\log 0.01 = 2

Remediation: Continue the pattern of the table backwards: log⁡100=2\log 100 = 2, log⁡10=1\log 10 = 1, log⁡1=0\log 1 = 0, log⁡0.1=?\log 0.1 = ?. Each division by 10 subtracts 1, so the next values are −1-1 and −2-2.

log⁡0=0\log 0 = 0

Remediation: Contrast log⁡1=0\log 1 = 0 (because 100=110^0 = 1) with 10n=010^n = 0, which has no solution. Let students try log(0) on the calculator and explain the error.

The log equals the number of digits

Remediation: Compare log⁡10=1\log 10 = 1 (2 digits) and log⁡999≈2.9996\log 999 \approx 2.9996 (3 digits). A dd-digit whole number has a log between d−1d - 1 and dd.

Differentiation Ideas

For Struggling Students:

  • • Start with a two-row table of powers of 10 and their logs, and let students fill it in from 10−310^{-3} to 10610^6
  • • Use only whole-number logs at first, then add one estimate like 1<log⁡50<21 < \log 50 < 2
  • • Let students count zeros and decimal places with a place value chart before writing the log

For On-Level Students:

  • • Estimate logs between whole numbers, then check them with the calculator
  • • Find the order of magnitude of quantities written in scientific notation
  • • Go back from a log to the number with 10log⁡x=x10^{\log x} = x

For Advanced Students:

  • • Explain why a whole number with dd digits has a log between d−1d - 1 and dd
  • • Compare log⁡2\log 2, log⁡20\log 20 and log⁡200\log 200 on the calculator and describe the pattern (it is explained in Product Rule for Logarithms)
  • • Research how log tables were used to multiply large numbers before calculators
Standards Alignment
  • HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)

    Express the solution to an exponential equation as a logarithm with base 2, 10 or e, and evaluate the logarithm using technology

  • HSF-BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)

    Understand the inverse relationship between exponents and logarithms

  • 8.EE.A.3 (CCSS.MATH.CONTENT.8.EE.A.3)

    Use numbers expressed as a single digit times an integer power of 10 to estimate very large or very small quantities

Lesson Resources
  • visualLog Scale Number Line

    Powers of 10 from 0.001 to 1 000 000 placed at equal steps, so each step adds 1 to the log

  • visualBar Chart of log x

    Compares the logs of 0.01, 0.1, 1, 5, 10, 50, 100 and 1000, including negative logs

  • activityEstimate, Then Press LOG

    Pairs write the two whole numbers around a log, then check with the calculator and score a point when the estimate holds

  • worksheetOrders of Magnitude Hunt

    Find real quantities (cell size, distances, populations) and order them by the whole-number part of their log

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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.

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

Why is base 10 called the common logarithm?

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.

How does the calculator find log⁡50\log 50?

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.

Why are logs of decimals like 0.010.01 negative?

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.

Is log⁡\log on my calculator the same as ln⁡\ln?

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 $10^{1.699} \approx 50$

More in This Topic

Introduction to Logarithms

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Definition of a Logarithm

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Natural Logarithms (ln)

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Graphing Logarithmic Functions

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