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

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Question 613455: How do you find log(x+21)+logx=2?
Found 2 solutions by radh, MathTherapy:
Answer by radh(108) About Me  (Show Source):
You can put this solution on YOUR website!
Combine using the product rule of logarithms.
log%28%28x%2B21%29%28x%29%29=2

Let both sides have a base of 10.
10%5E%28log%28x%28x%2B21%29%29%29=10%5E%282%29

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.
x%28x%2B21%29=10%5E%282%29

Multiply x by each term inside the parentheses.
x%5E%282%29%2B21x=10%5E%282%29

Square 10.
x%5E%282%29%2B21x=100

Move 100 to the other side.
x%5E%282%29%2B21x-100=0

Factor.

Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B21%2Ax%2B-100, 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: %28x%2Ba%29%28x%2Bb%29substitute 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 1%2Ax%5E2%2B21%2Ax%2B-100 again


Set each of the factors equal to 0.
x%2B25=0,x-4=0

Simplify.
x=-25,x=4

Here is the final answer:
x=-25,4

:)
Answer by MathTherapy(10858) About Me  (Show Source):

You can put this solution on YOUR website!
How do you find log(x+21)+logx=2?
*********************************
Respondent @radh(108) went along a very long path, and arrived at a PARTIALLY-CORRECT solution set. 

The smaller of the 2 logarithmic variable-arguments, x, MUST be > 0, so x > 0
We then get: log (x + 21) + log (x) = 2, with x > 0
  log x(x + 21) = 2 ---- Applying log+%28b%2C+%28c%29%29+%2B+log+%28b%2C+%28d%29%29 = log+%28b%2C+%28cd%29%29
log+%28%28x%5E2+%2B+21x%29%29+=+2
          x%5E2+%2B+21x+=+10%5E2 ---- Converting to EXPONENTIAL form
          x%5E2+%2B+21x+=+100
 x%5E2+%2B+21x+-+100+=+0
(x - 4)(x + 25) = 0 --- Factoring above TRINOMIAL
  x - 4 = 0     or       x + 25 = 0
        x = 4     or                x = - 25 (IGNORE)

The value, 4, for x, is > 0, but - 25 is NOT! Therefore, ONLY x = 4 is valid/ACCEPTABLE, while - 25 is an EXTRANEOUS solution.