Question 254819
problem is:


{{{log(2,x^2 + 4x + 3) - log(2,x^2 + x) = 4}}}


since {{{log(a,(b/c)) = log(a,b) - log(a,c)}}}, then the reverse is also true, i.e.:


{{{log(a,b) - log(a,c) = log(a,(b/c))}}}


applying this rule of logarithms makes your equation become:


{{{log(2,(x^2 + 4x + 3)/(x^2 + x)) = 4}}}


since {{{y = log(2,x)}}} if and only if {{{2^y = x}}}, applying this rule makes your equation become:


{{{2^4 = (x^2 + 4x + 3)/(x^2 + x)}}}


multiply both sides of this equation by {{{x^2 + x}}} and you get:


{{{2^4 * (x^2 + x)= (x^2 + 4x + 3)}}}


multiply the factors out and you get:


{{{16*x^2 + 16*x = x^2 + 4*x + 3}}}


subtract {{{x^2 + 4*x + 3}}} from both sides of this equation and combine like terms to get:


{{{15*x^2 + 12*x - 3 = 0}}}


factor this equation to get:


{{{15*x - 3) * (x + 1) = 0}}}


solve for x to get:


{{{x = (1/5)}}} or {{{x = -1}}}


confirm by substituting in original equation of:


{{{log(2,x^2 + 4x + 3) - log(2,x^2 + x) = 4}}}


to confirm using your calculator, you need to convert the base of the logarithm from 2 to 10 or e.


I used 10 because it is the LOG function of my calculator.


the conversion formula says:


{{{log(a,b) = log(c,b)/log(c,a)}}}


if we let a = base of 2 and we let c = base of 10, this formula becomes:


{{{log(2,b) = log(10,b)/log(10,2)}}}


your formula of {{{log(2,x^2 + 4x + 3) - log(2,x^2 + x) = 4}}} becomes:


{{{log(10,x^2 + 4x + 3)/log(10,2) - log(10,x^2 + x)/log(10,2) = 4}}}


substituting x = (1/5) into this equation, we get:


{{{log(10,(.2)^2 + 4(.2) + 3)/log(10,2) - log(10,(.2)^2 + (.2))/log(10,2) = 4}}}


this becomes:


{{{1.941106311 - (-2.058893689) = 4}}} which becomes:


{{{4 = 4}}} confirming that x = (1/5) is good.


when x = -1, confirmation was not successful because:


original equation is:


{{{log(2,x^2 + 4x + 3) - log(2,x^2 + x) = 4}}}


substitute -1 for x to get:


{{{log(2,(-1)^2 + (-1)*x + 3) - log(2,(-1)^2 + (-1)) = 4}}}


this becomes:


{{{log(2,0) - log(2,0) = 4}}}


since you can't take log of 0 to any base, x = -1 is not a valid answer.


graph of your original equation looks like this:


{{{graph(600,600,-4,2,-20,20,log(2,x^2 + 4x + 3) - log(2,x^2 + x) - 4)}}}


the answer to your problem is:


{{{x = (1/5)}}}