SOLUTION: log(3x^2+8)=1+log(x÷2+1)

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Question 1063951: log(3x^2+8)=1+log(x÷2+1)
Found 2 solutions by Edwin McCravy, MathTherapy:
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
log%28%283x%5E2%2B8%29%29=1%2Blog%28%28%28%22x%F72%22%29%5E%22%22%2B1%29%29

log%28%283x%5E2%2B8%29%29=1%2Blog%28%28%28x%2F2%29%5E%22%22%2B1%29%29

log%28%283x%5E2%2B8%29%29=1%2Blog%28%28%28x%2B2%29%2F2%29%29

log%28%283x%5E2%2B8%29%29=1%2Blog%28%28x%5E%22%22%2B2%29%29-log%28%282%5E%22%22%29%29

log%28%283x%5E2%2B8%29%29-log%28%28x%5E%22%22%2B2%29%29%2Blog%28%282%5E%22%22%29%29=1

log%28%28%283x%5E2%2B8%29%2F%28x%2B2%29%29%29%2Blog%28%282%5E%22%22%29%29=1

log%28%28%282%283x%5E2%2B8%29%29%2F%28x%5E%22%22%2B2%29%29%29=1

We use the fact that log(10) = 1, and that 10 is the
only such number,

%282%283x%5E2%2B8%29%29%2F%28x%2B2%29=10

Multiply both sides by (x+2)

2%283x%5E2%2B8%29=10%28x%2B2%29

Divide both sides by 2

3x%5E2%2B8=5%28x%2B2%29

3x%5E2%2B8=5x%2B10

3x%5E2-5x-2=0

%283x%2B1%29%28x-2%29=0

3x%2B1=0;   x-2=0
3x=-1;     x=2
x=-1%2F3

Both answers check.

Edwin

Answer by MathTherapy(10556) About Me  (Show Source):
You can put this solution on YOUR website!

log(3x^2+8)=1+log(x÷2+1)
log+%28%283x%5E2+%2B+8%29%29+=+1+%2B+log+%28%28%22x%F72%22+%2B+1%29%29
log+%28%283x%5E2+%2B+8%29%29+=+1+%2B+log+%28%28x%2F2+%2B+1%29%29
log+%28%283x%5E2+%2B+8%29%29+-+log+%28%28x%2F2+%2B+1%29%29+=+1
log+%28%28%283x%5E2+%2B+8%29%2F%28x%2F2+%2B+1%29%29%29+=+1
%283x%5E2+%2B+8%29%2F%28x%2F2+%2B+1%29+=+10%5E1
3x%5E2+%2B+8+=+10%28x%2F2+%2B+1%29 ------- Cross-multiplying
3x%5E2+%2B+8+=+5x+%2B+10
3x%5E2+-+5x+-+10+%2B+8+=+0
3x%5E2+-+5x+-+2+=+0
(x - 2)(3x + 1) = 0

highlight_green%28matrix%281%2C7%2C+x%2C+%22=%22%2C+2%2C+or%2C+x%2C+%22=%22%2C+-+1%2F3%29%29, since both values satisfy the equation