1.

The sum of all the solutions to the equation `2log_10x-log_10(2x-75)=2`a. 30b. 350c. 75d. 200

Answer» Given equation is ,
`2log_10x-log_10(2x-75) = 2`
Here, solution should satisfy two conditions.
(i) `x gt 0` and
(ii) `2x-75 gt 0 => x gt 75/2`
Now, we will solve the given equation,
`2log_10x-log_10(2x-75) = 2`
`=>log_10x^2-log_10(2x-75) = 2`
`=>log_10(x^2/(2x-75)) = 2`
`=>(x^2/(2x-75)) = 10^2`
`=>(x^2/(2x-75)) = 100`
`=> x^2-200x+7500 = 0`
`=>x^2-150x-50x+7500=0`
`=>(x-150)(x-50) = 0`
So, `x = 150` and `x = 50`
As both these values satisfying the above conditions (i) and (ii), both are the solutions.
`:. ` Sum of all solutions `= 150+50 = 200`


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