log x+log(x-48)=2

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violy's profile pic

violy | High School Teacher | (Level 1) Associate Educator

Posted on

Use the property: loga + logb = log(a*b). 

So, we will have: 

`log(x*(x-48)) = 2`

Use Distributive Property.

`log(x^2 - 48x) = 2`

Writing this in exponential form we will have:

`10^2 = x^2 - 48x`

`100 = x^2 - 48x` 

Use Completing the Square. 

Take the half of the coefficient of "x", then square the result. 

-48/2 = (-24)^2 = 576. 

Add 576 to both sides. 

`x^2 - 48x + 576 = 100 + 576`

`x^2 - 48x + 576 = 676`

Factor the left side. 

`(x - 24)^2 = 676`

Take the square root of both sides.

`x - 24 = +-26`

Solve for x on each case. 

`x - 24 = 26 ; x - 24 = - 26`

Add 24 on both sides.

`x = 50 ; x = -2`

We cannot have an answer which is negative, because of the logx. 

Hence, final answer here is x = 50. 

 

oldnick's profile pic

oldnick | (Level 1) Valedictorian

Posted on

`logx+log(x-48)=2`

`logx(x-48)=2`

`x(x-48)=100`

`x^2-48x-100=0`

`x^2-48x+576-100=576`

`x^2-48x+576=676`

`(x-24)^2=676`

`x-24=+-26`

`x=24+-26`

`x_1=50`  `x_2=-2`

`x_2=-2 `  notaccetable for function not defined in. 

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