document.write( "Question 613455: How do you find log(x+21)+logx=2? \n" ); document.write( "
Algebra.Com's Answer #386057 by radh(108)\"\" \"About 
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Combine using the product rule of logarithms.
\n" ); document.write( "\"log%28%28x%2B21%29%28x%29%29=2\"\r
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Let both sides have a base of 10.
\n" ); document.write( "\"10%5E%28log%28x%28x%2B21%29%29%29=10%5E%282%29\"\r
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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.
\n" ); document.write( "\"x%28x%2B21%29=10%5E%282%29\"\r
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Multiply x by each term inside the parentheses.
\n" ); document.write( "\"x%5E%282%29%2B21x=10%5E%282%29\"\r
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Square 10.
\n" ); document.write( "\"x%5E%282%29%2B21x=100\"\r
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Move 100 to the other side.
\n" ); document.write( "\"x%5E%282%29%2B21x-100=0\"\r
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Factor.
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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
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\n" ); document.write( " Factors:
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\n" ); document.write( " 1,2,4,5,10,20,25,50,100,
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\n" ); document.write( " -1,-2,-4,-5,-10,-20,-25,-50,-100,List the negative factors as well. This will allow us to find all possible combinations
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\n" ); document.write( " These factors pair up to multiply to -100.
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\n" ); document.write( " (-1)*(100)=-100
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\n" ); document.write( " (-2)*(50)=-100
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\n" ); document.write( " (-4)*(25)=-100
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\n" ); document.write( " (-5)*(20)=-100
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\n" ); document.write( " (-10)*(10)=-100
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\n" ); document.write( " 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
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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\r\n" ); document.write( " \r\n" ); document.write( " Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:\r\n" ); document.write( " \r\n" ); document.write( " \"%28x%2Ba%29%28x%2Bb%29\"substitute a=-4 and b=25\r\n" ); document.write( " \r\n" ); document.write( " So the equation becomes:\r\n" ); document.write( " \r\n" ); document.write( " (x-4)(x+25)\r\n" ); document.write( " \r\n" ); document.write( " Notice that if we foil (x-4)(x+25) we get the quadratic \"1%2Ax%5E2%2B21%2Ax%2B-100\" again\n" ); document.write( "

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Set each of the factors equal to 0.
\n" ); document.write( "\"x%2B25=0\",\"x-4=0\"\r
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Simplify.
\n" ); document.write( "\"x=-25\",\"x=4\"\r
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Here is the final answer:
\n" ); document.write( "x=-25,4\r
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:) \n" ); document.write( "

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