SOLUTION: Could someone please assist me by solving this logarithmic equation step-by-step for future reference. Thank you. log(7y + 1) = 2log(y+3)-log2

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Question 104689: Could someone please assist me by solving this logarithmic equation step-by-step for future reference. Thank you.
log(7y + 1) = 2log(y+3)-log2

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Some rules to remember
k+log+%28x%29=log%28%28x%5Ek%29%29
log%28%28x%29%29%2Blog%28%28y%29%29=log%28%28xy%29%29
log%28%28x%29%29-log%28%28y%29%29=log%28%28x%2Fy%29%29
Now let's look at your example.
log%28%287y+%2B+1%29%29+=+2log%28%28y%2B3%29%29-log%28%282%29%29
log%28%287y+%2B+1%29%29+-+2log%28%28y%2B3%29%29=+-log%28%282%29%29 Bring the variable expressions to one side.
log%28%287y+%2B+1%29%29+-+log%28%28y%2B3%29%5E2%29=+log%28%282%5E%28-1%29%29%29 Power rule from above both sides.
log%28%28%287y+%2B+1%29%2F%28y%2B3%29%5E2%29%29=+log%28%282%5E%28-1%29%29%29 Division rule from above.
%287y+%2B+1%29%2F%28y%2B3%29%5E2=+1%2F2 Inverse function both sides.
2%287y+%2B+1%29=%28y%2B3%29%5E2 Multiply both sides by 2%2A%28y%2B3%29%5E2
14y%2B2=y%5E2%2B6y%2B9 Distribute and expand.
y%5E2-8y%2B7=0Group all terms on one side.
%28y-1%29%28y-7%29=0 Factor the equation.
y=1 and y=7. Two solutions.
Verify your answer
log%28%287y+%2B+1%29%29+=+2log%28%28y%2B3%29%29-log%28%282%29%29
log%28%287%281%29+%2B+1%29%29+=+2log%28%281%2B3%29%29-log%28%282%29%29
log%28%288%29%29=2log%28%284%29%29-log%28%282%29%29
log%28%288%29%29=log%28%284%5E2%2F2%29%29 Power and division rule, right hand side.
log%28%288%29%29=log%28%288%29%29
Good answer.
log%28%287y+%2B+1%29%29+=+2log%28%28y%2B3%29%29-log%28%282%29%29
log%28%287%287%29+%2B+1%29%29+=+2log%28%287%2B3%29%29-log%28%282%29%29
log%28%2850%29%29=2log%28%2810%29%29-log%28%282%29%29
log%28%2850%29%29=log%28%2810%5E2%2F2%29%29 Power and division rule, right hand side.
log%28%2850%29%29=log%28%2850%29%29
Good answer.