SOLUTION: Solve the logarithmic equation. log x + log(x + 48) = 2

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Question 738480: Solve the logarithmic equation.
log x + log(x + 48) = 2
Found 2 solutions by josmiceli, MathLover1:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
The key is to make the substitution
+2+=+log%28+100+%29+
+log%28x%29+%2B+log%28+x+%2B+48+%29+=+log%28+100+%29+
+log%28+x%2A%28+x+%2B+48+%29+%29+=+log%28+100+%29+
+x%5E2+%2B+48x+-+100+=+0+
+%28+x+%2B+50+%29%2A%28+x+-+2+%29+=+0+ ( by inspection )
+x+=+-50+ ( disregard )
+x+=+2+
-----------
Check:
+log%28x%29+%2B+log%28+x+%2B+48+%29+=+log%28+100+%29+
+log%282%29+%2B+log%28+2+%2B+48+%29+=+log%28+100+%29+
+log%28+2%2A50+%29+=+log%28+100+%29+
+100+=+100+
OK
+x+=+-50+ cannot be an answer because the expession
+log%28x%29+ would make no sense. There cannot be a log
which will gives you a negative number.


Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
log+%28%28x%29%29+%2B+log%28%28x+%2B+48%29%29+=+2....in base 10 you will have

log+%28%28x%29%29%2Flog+%28%2810%29%29+%2B+log%28%28x+%2B+48%29%29%2Flog+%28%2810%29%29+=+2

%28log+%28%28x%29%29+%2B+log%28%28x+%2B+48%29%29%29%2Flog+%28%2810%29%29+=+2

log+%28%28x%29%29+%2B+log%28%28x+%2B+48%29%29+=+2log+%28%2810%29%29
log+%28x%28x+%2B+48%29%29+=+log+%28%2810%5E2%29%29
log+%28%28x%5E2+%2B+48x%29%29+=+log+%28%28100%29%29
x%5E2+%2B+48x+=+100
x%5E2+%2B+48x+-100=0...use quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-48+%2B-+sqrt%28+48%5E2-4%2A1%2A%28-100%29+%29%29%2F%282%2A1%29+
x+=+%28-48+%2B-+sqrt%28+2304%2B400+%29%29%2F2+
x+=+%28-48+%2B-+sqrt%28+2704+%29%29%2F2+
x+=+%28-48+%2B-+52%29%2F2+
solutions: we need only positive solution
x+=+%28-48+%2B+52%29%2F2+
x+=+4%2F2+
highlight%28x+=+2%29+