Logarithm Concepts -- Common Logarithms 1.2 2008/05/29 10:46:59 GMT-5 2008/12/30 15:45:49.929 US/Central Kenny Felder KFelder@RaleighCharterHS.org Kenny Felder KFelder@RaleighCharterHS.org algebra common felder logarithm This module covers some of the logarithms commonly encountered in algebra. When you see a root without a number in it, it is assumed to be a square root. That is, 25 size 12{ sqrt {"25"} } {}is a shorthand way of writing 252 size 12{ nroot { size 8{2} } {"25"} } {}. This rule is employed because square roots are more common than other types. When you see a logarithm without a number in it, it is assumed to be a base 10 logarithm. That is, log(1000) size 12{"log" \( "1000" \) } {} is a shorthand way of writing log10(1000) size 12{"log" rSub { size 8{"10"} } \( "1000" \) } {}. A base 10 logarithm is also known as a “common” log. Why are common logs particularly useful? Well, what is log10(1000) size 12{"log" rSub { size 8{"10"} } \( "1000" \) } {}? By now you know that this asks the question “10 to what power is 1000?” The answer is 3. Similarly, you can confirm that: log ( 10 ) = 1 size 12{"log" \( "10" \) =1} {} log ( 100 ) = 2 size 12{"log" \( "100" \) =2} {} log ( 1, 000 , 000 ) = 6 size 12{"log" \( 1,"000","000" \) =6} {} We can also follow this pattern backward: log ( 1 ) = 0 size 12{"log" \( 1 \) =0} {} log 1 10 = − 1 size 12{"log" left ( { {1} over {"10"} } right )= - 1} {} log 1 100 = − 2 size 12{"log" left ( { {1} over {"100"} } right )= - 2} {} and so on. In other words, the common log tells you the order of magnitude of a number: how many zeros it has. Of course, log10(500) size 12{"log" rSub { size 8{"10"} } \( "500" \) } {} is difficult to determine exactly without a calculator, but we can say immediately that it must be somewhere between 2 and 3, since 500 is between 100 and 1000.