SOLUTION: How do you find log(x+21)+logx=2?

Let both sides have a base of 10.

Remember that log is assumed to be 10 if there's no number specified. Because we made the equation have a base of 10, we can then cancel out the logs.

Multiply x by each term inside the parentheses.

Square 10.

Move 100 to the other side.

Factor.

Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)

In order to factor , first we need to ask ourselves: What two numbers multiply to -100 and add to 21? Lets find out by listing all of the possible factors of -100 Factors: 1,2,4,5,10,20,25,50,100, -1,-2,-4,-5,-10,-20,-25,-50,-100,List the negative factors as well. This will allow us to find all possible combinations These factors pair up to multiply to -100. (-1)*(100)=-100 (-2)*(50)=-100 (-4)*(25)=-100 (-5)*(20)=-100 (-10)*(10)=-100 Now which of these pairs add to 21? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 21 |||||||||| First Number | Second Number | Sum 1 | -100 | 1+(-100)=-99 2 | -50 | 2+(-50)=-48 4 | -25 | 4+(-25)=-21 5 | -20 | 5+(-20)=-15 10 | -10 | 10+(-10)=0 -1 | 100 | (-1)+100=99 -2 | 50 | (-2)+50=48 -4 | 25 | (-4)+25=21 -5 | 20 | (-5)+20=15 -10 | 10 | (-10)+10=0 We can see from the table that -4 and 25 add to 21.So the two numbers that multiply to -100 and add to 21 are: -4 and 25 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: substitute a=-4 and b=25 So the equation becomes: (x-4)(x+25) Notice that if we foil (x-4)(x+25) we get the quadratic again Set each of the factors equal to 0. , Simplify. , Here is the final answer: x=-25,4 :)