Question 1136697
a.)


log(11) = log(x^2 + 7)


this is true if and only if 11 = x^2 + 7


subtract 7 from both sides to get 4 = x^2.


solve for x to get x = plus or minus 2.


when x = plus or minus 2, x^2 is 4.


log(11) = log(x^2 + 7) becomes log(11) = log(4 + 7) which becomes log(11) = log(11) which confirms the solution is good.


b.)


e^(x^2-3) = e^(2x)


this is true if and only if x^2 - 3 = 2x


subtract 2x from both sides to get x^2 - 2x - 3 = 0


factor to get (x-3) * (x+1) = 0


solve for x to get x = -1 or x = 3.


when x = -1, e^(x^2-3) = e^(2x) becomes e^(1-3) = e^(2*-1) which becomes e^(-2) = e^(-2) which confirms the solution is good. 


when x = 3, e^(x^2-3) = e^(2x) becomes e^(9-3) = e^(2*3) which becomes e^(6) = e^(6) which confirms the solution is good.