Question

How do you solve `log x + log (x - 3) = 1`?

Answer 1

`x=5`

Explanation:

`logx+log(x-3)=1`
We know that: `loga+logb=log(a*b)`
`implies Log(x(x-3))=1`
`implies log(x^2-3x)=1`
`implies x^2-3x=10`
`implies x^2-3x-10=0`
`implies (x-5)(x+2)=0`
`implies x=5,-2`

Verification:-
Put `x=5`
`L.H.S=Logx+Log(x-3)=Log5+log(5-3)=log5+log2=log(5*2)=log10=1=R.H.S`
Verified.
Put `x=-2`
`L.H.S=Logx+Log(x-3)=Log(-2)+log(-2-3)=log(-2)+log(-5)`
Here we have to find the log of a negative number which is undefined.
Therefore not verified.

Therefore, only `x=5` is true.