Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Transcribed Image Text:Solve for x and check for any extraneous roots.
log (x + 15) + log (x) = 2
Expert Solution
Step 1
We have the following logarithm rules:
1) logc(a) + logc(b) = logc(ab)
2) logc(x) = y => x = cy
Given equation is:
log(x+15) + log(x) = 2
Apply the logarithm rule 1)
log[(x+15)x] = 2
=>log(x2+15x) = 2
Step 2
We have log(x2+15x) = 2
Apply the logarithm rule 2), here c = 10 as nothing is mentioned
=>x2+15x = 102
=>x2+15x-100 = 0
=>x2+20x-5x-100 = 0
=>x(x+20)-5(x+20) = 0
=>(x-5)(x+20) = 0
=>x-5 = 0; x+20 = 0
=>x = 5; x = -20
Check for extraneous roots by plugging these values in original equation
Check x = 5
log(5+15) + log(5) = 2
=>log(20) + log(5) = 2
=>log(20*5) = 2
=>log(100) = 2
=>100 = 102 , True
Check x = -20
log(-20+15) + log(-20) = 2
Here log (-x) is not defined and so log(-20) is not defined
=> x= -20 is extraneous solution
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