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.

Teacher Resources

Common Logarithms (Base 10)

Q: 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 $\log_{10} x$.

Q: 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.

Q: 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.

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 minLesson

Printable worksheet

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.

Try it now

What is

\log 10\,000

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 $d$ .

Interactive Visual

Base:

Calculate:10³ = 1000

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)

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

What is $\log 10\,000$ ?

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 $\log x$ 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

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

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 $10^{1.699} \approx 50$

Common Logarithms (Base 10)

Definition

Worked Examples

Common Mistakes

Interactive Visual

Bar Chart

Common logarithm of x

Interactive Sandbox

Expression Calculator

History

Practice Problems

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