Log Base 10 Calculator

What this calculator does

So, you want the logarithm of a number to base 10, the common logarithm. This tool takes a positive number and returns its log base 10, to five decimal places.

There is one input, the number, with the base fixed at 10. When people write simply "log" with no base shown, base 10 is very often what they mean, especially in science and engineering.

How to use it

1. Enter your number (x), which must be positive.
2. Press Calculate.

Log base 10: counting the tens

The log base 10 of a number is the power of 10 that produces it. Since 10 times 10 times 10 is 1000, the log base 10 of 1000 is 3. The log base 10 of 100 is 2, of 10 is 1, and of 1 is 0. Each time the number grows tenfold, its log goes up by exactly 1. That steady one-per-tenfold step is what makes base 10 so readable: the logarithm is basically counting how many factors of 10 are packed into the number.

Orders of magnitude and digits

Because of that, the common logarithm is really a measure of size, of what scientists call the order of magnitude. A number in the thousands has a log a little over 3, a number in the millions a log a little over 6. In fact, for a whole number, the log base 10 rounded down, plus 1, tells you how many digits it has: 4567 has a log of about 3.66, and 3 plus 1 is 4 digits. So the common logarithm turns the sprawling size of a number into a small, comparable figure, which is exactly why it underpins scientific notation.

The scales built on it

A lot of familiar measurement scales are common logarithms in disguise, precisely because base 10 tames huge ranges into a few tidy numbers. The pH scale in chemistry is the negative log base 10 of acidity, so each pH step means a tenfold change. The Richter scale for earthquakes is a log base 10 of amplitude, so a magnitude 6 quake shakes ten times harder than a magnitude 5. The decibel scale for sound works the same way. In every case, moving up by 1 on the scale means multiplying the underlying quantity by 10.

A worked example

Enter 1000. The tool returns 3, since 10 cubed is 1000. Enter 1 and it returns 0, because 10 to the power 0 is 1. Enter 0.01 and it returns minus 2, because 10 to the power minus 2 is one hundredth. Numbers below 1 give negative logs, and numbers above 1 give positive ones.

The logarithm of the tables

Base 10 has a special place in history. After John Napier introduced logarithms in the early 1600s, the English mathematician Henry Briggs proposed recasting them to base 10, with the log of 1 set to 0, precisely because that fitted the decimal number system everyone already used. The base 10 tables that followed, and the slide rules built on them, were the workhorses of calculation for over three centuries, until electronic calculators arrived. That is why base 10 logarithms are still called common logarithms today.

Questions people ask

What does log base 10 tell me?

The power of 10 that gives your number, which is how many factors of 10 it contains. The log of 1000 is 3.

How does it relate to the number of digits?

For a whole number, the log base 10 rounded down, plus 1, equals the number of digits. This is why it measures order of magnitude.

Why are pH and Richter based on it?

Because base 10 compresses vast ranges. Each step of 1 on those scales means a tenfold change in the underlying quantity.

Why can the answer be negative?

Numbers less than 1 are negative powers of 10, so their common logarithm is negative. The log of 0.01 is minus 2.

Why is it called the common logarithm?

Because base 10, championed by Henry Briggs, matched the decimal system and became the standard for calculation tables and slide rules for centuries.

References

On the common logarithm. The base 10 logarithm gives the power of 10 that produces a number, measuring its order of magnitude, and it was the base of the calculation tables.

1. Eric W. Weisstein, "Common Logarithm," from MathWorld, a Wolfram resource, on the logarithm to base 10.
2. "John Napier," MacTutor History of Mathematics, on Briggs proposing base 10 logarithms with the log of 1 set to 0.