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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 = 3 size 12{ sqrt {9} =3} {} can be rewritten as 3 2 = 9 size 12{3 rSup { size 8{2} } =9} {} . These two equations are the same statement about numbers, written in two different ways. 9 size 12{ sqrt {9} } {} asks the question “What number squared is 9?” So the equation 9 = 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 size 12{"log" rSub { size 8{3} } left ( { {1} over {3} } right )= - 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 1 3 size 12{ left ( { {1} over {3} } right )} {} ?” And the right side answers: “3 to the 1 size 12{ - 1} {} power is 1 3 size 12{ left ( { {1} over {3} } right )} {} .” 3 1 = 1 3 size 12{3 rSup { size 8{ - 1} } = left ( { {1} over {3} } right )} {} .

Log 3 of one-third equals -1

These two equations, log 3 1 3 = 1 size 12{"log" rSub { size 8{3} } left ( { {1} over {3} } right )= - 1} {} and 3 1 = 1 3 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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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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