How do you solve #log(x^2+4)-log(x+2)=2+log(x-2)#?
1) Domain of the logarithmic expressions
The first thing you need to do is establish the domain of your
#log(x^2 + 4)#is defined for #x^2 + 4 > 0#which is true for all #x in RR# #log(x+2)#is defined for #x + 2 > 0 <=> x > -2# #log(x-2)#is defined for #x - 2 > 0 <=> x > 2#
So, in total, the most restrictive condition is
2) Transform the equation and unite the logarithmic terms
Now, to "get rid" of the logarithmic terms, first of all, you need to eliminate the sums and unite your logarithmic terms.
The goal is to have just one
Use the logarithmic laws:
Now you can transform your equation as follows:
... use the logarithmic laws...
3) Eliminate the logarithmus
Now, you can "get rid" of the
The inverse function of
So, to eliminate the
4) Solve the quadratic equation
5) Determine the solution w.r.t. the domain
As we have established earlier that our domain is
Thus, we need to discard the negative solution, and the only solution in