Big Ideas Math A Bridge To Success Algebra 2: Student Edition 2015
1st Edition
ISBN: 9781680331165
Author: HOUGHTON MIFFLIN HARCOURT
Publisher: Houghton Mifflin Harcourt
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Question
Chapter 6.6, Problem 40E
To determine
To calculate: the value of x and check for extraneous solution.
Expert Solution & Answer
Answer to Problem 40E
The required real value of x are 2.72 .
Explanation of Solution
Given Information:
The given equation is log5(x+4)+log5(x+1)=2 .
Formula Used:
Product rule of logarithm:
logax+logay=loga(xy)
Power rule of logarithm:
logaxp=plogax
Property of logarithms:
If loga(x)=b then x=ab
If ax2+bx+c=0 then x=−b±√b2−4ac2a .
Where a,b,c are constant and a≠0 .
Calculation:
Consider the equation is log5(x+4)+log5(x+1)=2 .
Use product rule of logarithm logax+logay=loga(xy) in the above equation.
log5(x+4)(x+1)=2
Now, use the property of logarithms of the above equation.
(x+4)(x+1)=52
Simplify the above equation.
(x+4)(x+1)=52x2+x+4x+4=25x2+5x+4−25=0x2+5x−21=0
Now, use quadratic formula in the above equation x2+5x−21=0 .
x=−b±√b2−4ac2a
Substitute 1 for a , 5 for b and −21 for c in the above equation.
x=−5±√52−4(1)(−21)2(1)
Simplify the above equation.
x=−5±√52−4(1)(−21)2(1)=−5±√25+842=−5±√1092=−5±10.442
Simplify the equation.
x=−5±10.442=−5+10.442,−5−10.442=5.442,−15.442=2.72,−7.72
Check for extraneous solution.
For x=2.72 .
log5(x+4)+log5(x+1)=2
Substitute 2.72 for x in the given above equation.
log5(x+4)+log5(x+1)=2
Simplify the above equation.
log5(x+4)+log5(x+1)=2log5(2.72+4)+log5(2.72+1)=2log5(6.72)+log5(6.72)=2
Substitute 1.18 for log5(6.72) and 0.82 for log5(6.72) in the above equation.
1.18+0.82=22=2
For x=−7.72 .
log5(x+4)+log5(x+1)=2
Substitute −7.72 for x in the given above equation.
log5(x+4)+log5(x+1)=2
Simplify the above equation.
log5(x+4)+log5(x+1)=2log5(−7.72+4)+log5(−7.72+1)=2log5(−3.72)+log5(−6.72)=2
It is known as loga(−x) is does not exist.
Thus, at x=−7.72 the given equation has extraneous solution.
Hence, the required real values of x are 2.72 .
Chapter 6 Solutions
Big Ideas Math A Bridge To Success Algebra 2: Student Edition 2015
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