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6.1 The logarithm explained by analogy to roots

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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. 8 3 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 25 size 12{ sqrt {"25"} } {} , you just have to know that 5 2 = 25 size 12{5 rSup { size 8{2} } ="25"} {} . If someone asks for 30 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.

Roots Logs
The question x a size 12{ nroot { size 8{a} } {x} } {} means “what number, raised to the a power, is x?” As an equation, ? a = x size 12{? rSup { size 8{a} } =x} {} log a x size 12{"log" rSub { size 8{a} } x} {} means “ a size 12{a} {} , raised to what power, is x size 12{x} {} ?” As an equation, a ? = x size 12{a rSup { size 8{?} } =x} {}
Example that comes out even 8 3 = 2 size 12{ nroot { size 8{3} } {8} =2} {} log 2 8 = 3 size 12{"log" rSub { size 8{2} } 8=3} {}
Example that doesn’t 10 3 size 12{ nroot { size 8{3} } {"10"} } {} is a bit more than 2 log 2 10 size 12{"log" rSub { size 8{2} } "10"} {} is a bit more than 3
Out of domain example − 4 size 12{ sqrt { - 4} } {} does not exist ( x 2 size 12{x rSup { size 8{2} } } {} will never give − 4 size 12{ - 4} {} ) log 2 ( 0 ) size 12{"log" rSub { size 8{2} } \( 0 \) } {} and log 2 ( − 1 ) size 12{"log" rSub { size 8{2} } \( - 1 \) } {} do not exist ( 2 x size 12{2 rSup { size 8{x} } } {} will never give 0 or a negative answer)
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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