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Solve for x. log(base 5)(x + 22) - log(base 5)(x - 2) = 2 If there are multiple...

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kristenmarieb... | Student, Grade 10 | Valedictorian

Posted August 24, 2013 at 4:24 AM via web

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Solve for x.

log(base 5)(x + 22) - log(base 5)(x - 2) = 2

If there are multiple answers, separate with a comma.

Tagged with algebra2, math

2 Answers | Add Yours

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted August 24, 2013 at 4:31 AM (Answer #1)

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The equation `log_5(x + 22) - log_5(x - 2) = 2` has to be solved for x.

`log_5(x + 22) - log_5(x - 2) = 2`

= `log_5((x+22)/(x - 2)) = 2`

=> `(x+22)/(x - 2) = 5^2`

=> `(x+22)/(x - 2) = 25`

=> x + 22 = 25x - 50

=> 24x = 72

=> x = 3

The solution of the equation is x = 3

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baxthum8 | High School Teacher | (Level 3) Associate Educator

Posted August 24, 2013 at 9:54 AM (Answer #2)

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In order to solve `log_(5)(x+22) - log_(5)(x - 2) = 2`

we will use one of the properties of logarithms:  

  `log_(b)x - log_(b)y = log_(b)(x/y)`

Therefore we will have:

`log_(5)(x+22) - log_(5)(x - 2) = 2 rArr log_(5) ( x+22 ) / ( x-2 ) = 2` 

Using the meaning of logs, this equation equals:

`5^2 = ( x+22 ) / ( x - 2 )`     `rArr`     `25 = ( x+22 ) / ( x - 2 )` We will now multiply both sides by ( x - 2) to get:

`25 ( x - 2 ) = x + 22` Distribute the 25.

`25x - 50 = x + 22` Subtract x from both sides.

`24x - 50 = 22` Add 50 to both sides.

`24x = 72` Divide both sides by 24.

The final answer is `x = 3.`

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