Function graph of the decadic logarithm

The decadic logarithm or the logarithm of ten is the base 10 logarithm . The mathematical notation for the decadic logarithm of a number is in accordance with DIN 1302${\ displaystyle x}$

${\ displaystyle \ lg x}$ or ${\ displaystyle \ log _ {10} x \ ,.}$

Its inverse function is , that is, is synonymous with${\ displaystyle 10 ^ {x}}$${\ displaystyle y = 10 ^ {x}}$${\ displaystyle x = \ lg y \ ,.}$

The notation (without a base) has contradicting meanings (see logarithm ), but in practice it is sometimes used for the decadic logarithm. ${\ displaystyle \ log x}$

Log tables made arithmetic easier before pocket calculators became a widespread tool in the 1970s . In Annexes many books there were tables of logarithms , the listed.If the value of the logarithm of all numbers from 1 to 10 in steps of, for example 0.01 or 0.001. Only the values ​​for numbers from 1 to 10 had to be printed, since the values ​​for other numbers can be calculated as in the following example. If you read from the table that

${\ displaystyle \ log _ {10} 1 {,} 2 \ approx 0 {,} 07918}$

holds, it follows

${\ displaystyle \ log _ {10} 120 = \ log _ {10} (10 ^ {2} \ cdot 1 {,} 2) = 2 + \ log _ {10} 1 {,} 2 \ approx 2 {, } 07918 \ ,.}$

The decadic logarithm is also called Briggs' logarithm after Henry Briggs .

## Base conversion

Today, many scientific calculators (such as those used in schools) have a button labeled logthat represents the logarithm of a number. If you want to get the logarithm on the basis of another number and only have one key available for the logarithm on the base 10, the following mathematical law can help you:

${\ displaystyle \ log _ {b} (x) = {\ frac {\ log _ {a} (x)} {\ log _ {a} (b)}}}$

### example

In this calculation example, the binary logarithm is calculated using the decadic logarithm: ${\ displaystyle \ log _ {2} (16)}$

${\ displaystyle \ log _ {2} (16) = {\ frac {\ lg (16)} {\ lg (2)}} = {\ frac {1 {,} 2041 \ ldots} {0 {,} 3010 \ ldots}} = 4}$