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

For Teachers

Learning Objectives

Read $\log x$ as the logarithm with base 10
Evaluate $\log(10^n) = n$ for powers of 10 from $0.001$ to $1\,000\,000$ 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 $\log x$ to scientific notation and orders of magnitude

Prerequisites

• Introduction to Logarithms and Definition of a Logarithm: $\log_b y = x \iff b^x = y$
• Powers of 10, including negative exponents ( $10^{-2} = 0.01$ )
• Scientific notation ( $3\,000 = 3 \times 10^3$ )

Discussion Starters

1. Why do you think base 10 logs were the first to be printed in tables and put on calculators?
2. If $\log x$ goes up by 1, what happens to $x$ ? And if it goes up by 2?
3. $50$ is about halfway between $10$ and $100$ . Why is $\log 50$ not $1.5$ ?
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 $\log 50 = 1.5$

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

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

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

$\log 0 = 0$

Remediation: Contrast $\log 1 = 0$ (because $10^0 = 1$ ) with $10^n = 0$ , 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 $\log 10 = 1$ (2 digits) and $\log 999 \approx 2.9996$ (3 digits). A $d$ -digit whole number has a log between $d - 1$ and $d$ .

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 $10^{-3}$ to $10^6$
• Use only whole-number logs at first, then add one estimate like $1 < \log 50 < 2$
• 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 $10^{\log x} = x$

For Advanced Students:

• Explain why a whole number with $d$ digits has a log between $d - 1$ and $d$
• Compare $\log 2$ , $\log 20$ and $\log 200$ 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_{10} x

\log x

answers the question: **"10 to which power gives

x

?"**

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

Powers of 10 give whole-number logs:

$x$	$0.001$	$0.01$	$0.1$	$1$	$10$	$100$	$1\,000$	$1\,000\,000$
$\log x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$	$6$

\log(10^n) = n

for every exponent

n

Numbers between two powers of 10 have a log between two whole numbers. Since

10 < 50 < 100

, we know

1 < \log 50 < 2

. A calculator (the LOG key) gives

\log 50 \approx 1.699

The argument must be positive:

\log 0

and

\log(-5)

do not exist.

Worked Examples

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

Ask the question for 100 000

10 to which power gives $100\,000$ ? → $10^n = 100\,000$

Count the zeros

$100\,000$ is a 1 followed by 5 zeros, so $100\,000 = 10^5$ → $\log 100\,000 = 5$

Ask the question for 1

Any nonzero number to the power 0 is 1: $10^0 = 1$ → $\log 1 = 0$

Answer: $\log 100\,000 = 5$ and $\log 1 = 0$

Common Mistakes

Thinking $\log 50 = 1.5$ 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 $10^{1.5} \approx 31.6$ .

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

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

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

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

Writing $\log 0 = 0$ or $\log(-100) = -2$

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

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

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

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

Correct: $\log 1\,000 = 3$ because $10^3 = 1\,000$ . A whole number with $d$ digits has a log between $d - 1$ and $d$ .

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 $\log x$ is the exponent of $x$ in scientific notation. $\log(3\,000) \approx 3.477$ and $3\,000 = 3 \times 10^3$ .
Chemistry: the pH of a solution is $-\log$ of its hydrogen ion concentration.
Sound: decibels use $10 \log$ 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 $1$ , $10$ , $100$ and $1\,000\,000$ 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 $0.0001$ m thick, and $\log 0.0001 = -4$ . The Earth is about $10\,000\,000$ m across, and $\log 10\,000\,000 = 7$ . The difference of 11 orders of magnitude means the Earth is about $10^{11}$ times wider.

1Try It Yourself

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

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

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

2Try It Yourself

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

What is its pH?

Step 1: Write the mathematical expression

Calculate $-\log 0.0001$ :

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 $100$ times the threshold: $L = 10 \log 100 = 10 \times 2 = 20$ dB.

3Try It Yourself

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

What is its sound level in decibels?

Step 1: Write the mathematical expression

Calculate $10 \log 1\,000\,000$ :

Key Takeaways

1The common logarithm has base 10: $\log x = \log_{10} x$ , and $\log x = n$ means $10^n = x$
2Powers of 10 have whole-number logs: $\log(10^n) = n$ , for example $\log 1\,000 = 3$ and $\log 0.01 = -2$
3 $\log 1 = 0$ and $\log 10 = 1$ ; numbers between 0 and 1 have negative logs
4To estimate $\log x$ , find the powers of 10 around $x$ : $10 < 50 < 100$ gives $1 < \log 50 < 2$
5The whole-number part of $\log x$ is the order of magnitude, the exponent of $x$ in scientific notation
6Use the LOG key for exact decimals ( $\log 50 \approx 1.699$ ); $\log 0$ 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

means

\log_{10} x

How does the calculator find $\log 50$ ?

It uses a fast numeric method to find the exponent

n

with

10^n = 50

. You can check its answer:

10^{1.699} \approx 50

. The estimate

1 < \log 50 < 2

tells you if you pressed the right keys.

Why are logs of decimals like $0.01$ negative?

Numbers between 0 and 1 are reached with negative exponents:

0.01 = \dfrac{1}{100} = 10^{-2}

. So

\log 0.01 = -2

. Only the argument must be positive, the log itself can be any real number.

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

No. LOG uses base 10, while LN uses a different base, the number

e \approx 2.718

. For example

\log 100 = 2

, but

\ln 100 \approx 4.605

. The lesson Natural Logarithms (ln) covers that key.

Glossary

Common logarithm: The logarithm with base 10, written $\log x$ : the exponent $n$ with $10^n = x$
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 $x$ in $\log x$ ; it must be positive
Power of 10: A number of the form $10^n$ , such as $1\,000 = 10^3$ or $0.01 = 10^{-2}$
Order of magnitude: The whole-number part of $\log x$ , which is the exponent of $x$ in scientific notation
Scientific notation: Writing a number as $a \times 10^n$ with $1 \le a < 10$ , such as $3\,000 = 3 \times 10^3$

Formula Card

Common logarithm

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

The base 10 is not written

Log of a power of 10

\log(10^n) = n

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

Special values

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

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

Estimating

10^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

10^{\log x} = x

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

Teacher Guide: Common Logarithms (Base 10)

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Orders of Magnitude

pH in Chemistry

Decibels and Sound

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Common logarithm

Log of a power of 10

Special values

Estimating

Back to the number

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Natural Logarithms (ln)

Graphing Logarithmic Functions