Question

How do you solve `log (x – 2) + log x = log 3`?

Answer 1

This equation has one solution `x=3`

Explanation:

We start with: `log(x-2)+logx=log3`

Before calculating `x` we have to find the domain of the equation.
Since `log` is only defined for positive values we have to find out where `x-2>0`, so we get the domain: `D=(2;+oo)`

`logx*(x-2)=log3`

`x*(x-2)=3`

`x^2-2x=3`

`x^2-2x-3=0`

`Delta=4-4*1*(-3)=4+12=16`

`sqrt(Delta)=4`

`x_1=(2-4)/2=-1`

`x_2=(2+4)/2=3`

From those values only `x_2` is in the domain `D` so it is the only solution of the equation.

Answer: `x=2`