SOLUTION: {{{In ^4sqrt(ab^2)}}} I do not know how to express this in terms of sums and differences of logarithms. Please help. Thank you very very much.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: {{{In ^4sqrt(ab^2)}}} I do not know how to express this in terms of sums and differences of logarithms. Please help. Thank you very very much.       Log On


   

Question 153014: In+%5E4sqrt%28ab%5E2%29 I do not know how to express this in terms of sums and differences of logarithms. Please help. Thank you very very much.
Found 2 solutions by Fombitz, nabla:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!

Two logarithm rules are needed here,
ln%28X%29%2Bln%28Y%29=ln%28XY%29
a%2Aln%28Z%29=ln%28Z%5Ea%29
.
.
.
ln%28%28ab%5E2%29%5E%281%2F4%29%29=ln%28a%5E%281%2F4%29%2Ab%5E%281%2F2%29%29
ln%28%28ab%5E2%29%5E%281%2F4%29%29=ln%28a%5E%281%2F4%29%29%2Bln%28b%5E%281%2F2%29%29
ln%28%28ab%5E2%29%5E%281%2F4%29%29=%281%2F4%29%2Aln%28a%29%2B%281%2F2%29%2Aln%28b%29

Answer by nabla(475) About Me  (Show Source):
You can put this solution on YOUR website!
4th root is the same as to the 1/4 power. We take powers out of logarithms as fraction coefficients.
So In+%5E4sqrt%28ab%5E2%29=1/4(ln ab^2)
Now, multiplied variables/numbers in logarithms can be taken out as addition.
So (1/4)(ln ab^2)=(1/4)(ln a)+(1/4)(ln b^2)
Now, repeat the first step for the logarithm with b^2.
So (1/4)(ln a)+(1/2)(ln b)