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The Logarithm Defined as an Inverse Function

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

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

xx size 12{ sqrt {x} } {} can be defined as the inverse function of x2x2 size 12{x rSup { size 8{2} } } {}. Recall the definition of an inverse function— f1(x)f1(x) size 12{f rSup { size 8{ - 1} } \( x \) } {} is defined as the inverse of f1(x)f1(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:

Table 1
xx size 12{ sqrt {x} } {} is the inverse function of x2x2 size 12{x rSup { size 8{2} } } {}
3 x 2 9 3 x 2 9 size 12{3 rightarrow x rSup { size 8{2} } rightarrow 9} {}
9 x 3 9 x 3 size 12{9 rightarrow sqrt {x} rightarrow 3} {}

Similarly, log2xlog2x size 12{"log" rSub { size 8{2} } x} {} is the inverse function of the exponential function 2x2x size 12{2 rSup { size 8{x} } } {}.

Table 2
log2xlog2x size 12{"log" rSub { size 8{2} } x} {} is the inverse function of 2x2x size 12{2 rSup { size 8{x} } } {}
3 2 x 8 3 2 x 8 size 12{3 rightarrow 2 rSup { size 8{x} } rightarrow 8} {}
8 log 2 x 2 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, 2x2x 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, xx size 12{ sqrt {x} } {}is not a perfect inverse of x2x2 size 12{x rSup { size 8{2} } } {}, since it does not work for negative numbers. (3)2=9(3)2=9 size 12{ \( - 3 \) rSup { size 8{2} } =9} {}, but 99 size 12{ sqrt {9} } {} is not 33 size 12{ - 3} {}. Logarithms have no such limitation: log2xlog2x size 12{"log" rSub { size 8{2} } x} {} is a perfect inverse for 2x2x 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. 3232 size 12{3 rSup { size 8{2} } } {} and 2323 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 1010 size 12{ sqrt {"10"} } {} and log210log210 size 12{"log" rSub { size 8{2} } "10"} {}, respectively, and they have different answers. x2x2 size 12{x rSup { size 8{2} } } {} and 2x2x size 12{2 rSup { size 8{x} } } {} are not the same function, and they therefore have different inverse functions xx size 12{ sqrt {x} } {} and log210log210 size 12{"log" rSub { size 8{2} } "10"} {}.

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