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Rewriting Logarithm Equations as Exponent Equations

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

Summary: This module discusses how equations involving logarithms can be re-written using exponents.

Both root equations and logarithm equations can be rewritten as exponent equations.

9=39=3 size 12{ sqrt {9} =3} {} can be rewritten as 32=932=9 size 12{3 rSup { size 8{2} } =9} {}. These two equations are the same statement about numbers, written in two different ways. 99 size 12{ sqrt {9} } {} asks the question “What number squared is 9?” So the equation 9=39=3 size 12{ sqrt {9} =3} {}asks this question, and then answers it: “3 squared is 9.”

We can rewrite logarithm equations in a similar way. Consider this equation:

log 3 1 3 = 1 log 3 1 3 = 1 size 12{"log" rSub { size 8{3} } left ( { {1} over {3} } right )= - 1} {}
(1)

If you are asked to rewrite that logarithm equation as an exponent equation, think about it this way. The left side asks: “3 to what power is 1313 size 12{ left ( { {1} over {3} } right )} {}?” And the right side answers: “3 to the 11 size 12{ - 1} {}power is 1313 size 12{ left ( { {1} over {3} } right )} {}.” 31=1331=13 size 12{3 rSup { size 8{ - 1} } = left ( { {1} over {3} } right )} {}.

Figure 1
Log 3 of one-third equals -1

These two equations, log313=1log313=1 size 12{"log" rSub { size 8{3} } left ( { {1} over {3} } right )= - 1} {}and 31=1331=13 size 12{3 rSup { size 8{ - 1} } = left ( { {1} over {3} } right )} {}, are two different ways of expressing the same numerical relationship.

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