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Logarithm Concepts -- The logarithm explained by analogy to roots

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

Summary: This module discusses another way of understanding logarithms by providing an analogy to roots.

The logarithm may be the first really new concept you’ve encountered in Algebra II. So one of the easiest ways to understand it is by comparison with a familiar concept: roots.

Suppose someone asked you: “Exactly what does root mean?” You do understand roots, but they are difficult to define. After a few moments, you might come up with a definition very similar to the “question” definition of logarithms given above. 8383 size 12{ nroot { size 8{3} } {8} } {} means “what number cubed is 8?”

Now the person asks: “How do you find roots?” Well...you just play around with numbers until you find one that works. If someone asks for 2525 size 12{ sqrt {"25"} } {}, you just have to know that 52=2552=25 size 12{5 rSup { size 8{2} } ="25"} {}. If someone asks for 3030 size 12{ sqrt {"30"} } {}, you know that has to be bigger than 5 and smaller than 6; if you need more accuracy, it’s time for a calculator.

All that information about roots applies in a very analogous way to logarithms.

Table 1
  Roots Logs
The question xaxa size 12{ nroot { size 8{a} } {x} } {} means “what number, raised to the a power, is x?” As an equation, ?a=x?a=x size 12{? rSup { size 8{a} } =x} {} logaxlogax size 12{"log" rSub { size 8{a} } x} {} means “ aa size 12{a} {}, raised to what power, is xx size 12{x} {}?” As an equation, a?=xa?=x size 12{a rSup { size 8{?} } =x} {}
Example that comes out even 8 3 = 2 8 3 = 2 size 12{ nroot { size 8{3} } {8} =2} {} log 2 8 = 3 log 2 8 = 3 size 12{"log" rSub { size 8{2} } 8=3} {}
Example that doesn’t 103103 size 12{ nroot { size 8{3} } {"10"} } {} is a bit more than 2 log210log210 size 12{"log" rSub { size 8{2} } "10"} {} is a bit more than 3
Out of domain example −4−4 size 12{ sqrt { - 4} } {}does not exist ( x2x2 size 12{x rSup { size 8{2} } } {} will never give −4−4 size 12{ - 4} {}) log2(0)log2(0) size 12{"log" rSub { size 8{2} } \( 0 \) } {} and log2(−1)log2(−1) size 12{"log" rSub { size 8{2} } \( - 1 \) } {} do not exist ( 2x2x size 12{2 rSup { size 8{x} } } {} will never give 0 or a negative answer)

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