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Logarithm Concepts -- Common Logarithms

Module by: Kenny M. Felder. E-mail the author

Summary: 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, 2525 size 12{ sqrt {"25"} } {}is a shorthand way of writing 252252 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)log(1000) size 12{"log" \( "1000" \) } {} is a shorthand way of writing log10(1000)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)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 log ( 10 ) = 1 size 12{"log" \( "10" \) =1} {}
(1)
log ( 100 ) = 2 log ( 100 ) = 2 size 12{"log" \( "100" \) =2} {}
(2)
log ( 1, 000 , 000 ) = 6 log ( 1, 000 , 000 ) = 6 size 12{"log" \( 1,"000","000" \) =6} {}
(3)

We can also follow this pattern backward:

log ( 1 ) = 0 log ( 1 ) = 0 size 12{"log" \( 1 \) =0} {}
(4)
log 1 10 = − 1 log 1 10 = − 1 size 12{"log" left ( { {1} over {"10"} } right )= - 1} {}
(5)
log 1 100 = − 2 log 1 100 = − 2 size 12{"log" left ( { {1} over {"100"} } right )= - 2} {}
(6)

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)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.

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