Question 240493
You have unbalanced parentheses in your problem so I have to check. Is this the original logarithm?
{{{ln((3x^2)/sqrt(2x+1))}}}
If yes, keep reading. If not, then please repost your problem with properly balanced parentheses.<br>
Your answer, {{{2ln(3x) + (1/2)ln(2x+1)}}}, is pretty close. One error is that it should be a difference instead of the sum because of the division in the original log:
{{{ln((3x^2)/sqrt(2x+1)) = 2ln(3x) - (1/2)ln(2x+1)}}}
This may meet the requirements of the problem. But it is possible to use  the property, {{{log(a, (p*q)) = log(a, (p)) + log(a, (q))}}}, on the first log giving:
{{{ln((3x^2)/sqrt(2x+1)) = 2ln(3x) - (1/2)ln(2x+1) = 2(ln(3) + ln(x)) - (1/2)ln(2x+1) = 2ln(3) + 2ln(x) - (1/2)ln(2x+1)}}}
I don't know if this last expression would be preferred over {{{2ln(3x) - (1/2)ln(2x+1)}}}.