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What is x in equation lgx+lg(x+90)=3?

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greenbel

Posted June 28, 2013 at 1:55 PM via web

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What is x in equation lgx+lg(x+90)=3?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted June 28, 2013 at 2:10 PM (Answer #1)

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You need to convert the sum of logarithms into the logarithm of product, such that:

`log x + log(x + 90) = log (x(x + 90))`

Replacing `log (x(x + 90))` for `log x + log(x + 90)` in equation, yields:

`log (x(x + 90)) = 3`

Since the base of logarithm is 10 yields:

`(x(x + 90)) = 10^3 => x^2 + 90x - 1000 = 0`

You need to use quadratic formula, such that:

`x_(1,2) = (-90+-sqrt(8100 + 4000))/2 => x_(1,2) = (-90+-110)/2`

`x_1 = 10 ; x_2 = -100`

Testing the solutions in equation yields:

`log 10 + log(10 + 90) = 1 + log 100 = 1 + log 10^2 = 1 + 2log 10 = 1 + 2 = 3`

`log -100 + log (-100 + 90)` invalid

Hence, evaluating the solution to the given equation yields `x = 10.`

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