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Solve the following logarithmic equation: `log_2(3x-2) - log_2 (x-5) = 4`

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nanalisa | Student, Undergraduate | Honors

Posted February 3, 2012 at 10:37 PM via web

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Solve the following logarithmic equation: `log_2(3x-2) - log_2 (x-5) = 4`

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted February 3, 2012 at 10:58 PM (Answer #1)

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The equation to be solved is `log_2 (3x - 2) - log_2 (x - 5) = 4`

Use the property of logarithm log a - log b = log(a/b)

`log_2 (3x - 2) - log_2 (x - 5) = 4`

=> `log_2 ((3x - 2)/(x - 5)) = 4`

=> `(3x - 2)/(x - 5) = 2^4`

=> `(3x - 2)/(x - 5) = 16`

=> 3x - 2 = 16x - 80

=> 13x = 78

=> x = 6

The solution of the equation is x = 6

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted February 3, 2012 at 10:59 PM (Answer #2)

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Notice that bases of logarithms are alike, hence, you need to transform the difference into a quotient such that:

`log_2((3x-2)/(x-5)) = 4 =gt ((3x-2)/(x-5)) = 2^4` `3x - 2 = 16(x-5)` 

Opening the brackets yields:

3x - 2 = 16x - 80 => -13x = -78 => x = 6

Hence the solution to the equation is x = 6.

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