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This module discusses how logarithms can be defined as inverse functions.

x size 12{ sqrt {x} } {} can be defined as the inverse function of x 2 size 12{x rSup { size 8{2} } } {} . Recall the definition of an inverse function— f 1 ( x ) size 12{f rSup { size 8{ - 1} } \( x \) } {} is defined as the inverse of f 1 ( x ) size 12{f rSup { size 8{1} } \( x \) } {} if it reverses the inputs and outputs. So we can demonstrate this inverse relationship as follows:

x size 12{ sqrt {x} } {} is the inverse function of x 2 size 12{x rSup { size 8{2} } } {}
3 x 2 9 size 12{3 rightarrow x rSup { size 8{2} } rightarrow 9} {}
9 x 3 size 12{9 rightarrow sqrt {x} rightarrow 3} {}

Similarly, log 2 x size 12{"log" rSub { size 8{2} } x} {} is the inverse function of the exponential function 2 x size 12{2 rSup { size 8{x} } } {} .

log 2 x size 12{"log" rSub { size 8{2} } x} {} is the inverse function of 2 x size 12{2 rSup { size 8{x} } } {}
3 2 x 8 size 12{3 rightarrow 2 rSup { size 8{x} } rightarrow 8} {}
8 log 2 x 2 size 12{8 rightarrow "log" rSub { size 8{2} } x rightarrow 2} {}

(You may recall that during the discussion of inverse functions, 2 x size 12{2 rSup { size 8{x} } } {} was the only function you were given that you could not find the inverse of. Now you know!)

In fact, as we noted in the first chapter, x size 12{ sqrt {x} } {} is not a perfect inverse of x 2 size 12{x rSup { size 8{2} } } {} , since it does not work for negative numbers. ( 3 ) 2 = 9 size 12{ \( - 3 \) rSup { size 8{2} } =9} {} , but 9 size 12{ sqrt {9} } {} is not 3 size 12{ - 3} {} . Logarithms have no such limitation: log 2 x size 12{"log" rSub { size 8{2} } x} {} is a perfect inverse for 2 x size 12{2 rSup { size 8{x} } } {} .

The inverse of addition is subtraction. The inverse of multiplication is division. Why do exponents have two completely different kinds of inverses, roots and logarithms? Because exponents do not commute . 3 2 size 12{3 rSup { size 8{2} } } {} and 2 3 size 12{2 rSup { size 8{3} } } {} are not the same number. So the question “what number squared equals 10?” and the question “2 to what power equals 10?” are different questions, which we express as 10 size 12{ sqrt {"10"} } {} and log 2 10 size 12{"log" rSub { size 8{2} } "10"} {} , respectively, and they have different answers. x 2 size 12{x rSup { size 8{2} } } {} and 2 x size 12{2 rSup { size 8{x} } } {} are not the same function, and they therefore have different inverse functions x size 12{ sqrt {x} } {} and log 2 10 size 12{"log" rSub { size 8{2} } "10"} {} .

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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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