SOLUTION: Write the expression as a single logarithm. https://i.ibb.co/m9Wr8vy/1.jpg The answer is here https://i.ibb.co/gwMRMC7/2.jpg but I can't figure out the steps

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Write the expression as a single logarithm. https://i.ibb.co/m9Wr8vy/1.jpg The answer is here https://i.ibb.co/gwMRMC7/2.jpg but I can't figure out the steps      Log On


   

Question 1204581: Write the expression as a single logarithm.
https://i.ibb.co/m9Wr8vy/1.jpg
The answer is here https://i.ibb.co/gwMRMC7/2.jpg
but I can't figure out the steps
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


For simplicity, we can ignore the log base 7 and consider the same expression in standard base 10 logarithms.

log%28%28root%283%2Cx%29%29%29-log%28%28y%5E3%29%29%2B2log%28%28y%29%29

Write the expression as a single logarithm using 3 basic log rules:
(1) a%2Alog%28%28b%29%29=log%28%28b%5Ea%29%29
(2) log%28%28a%29%29-log%28%28b%29%29=log%28%28a%2Fb%29%29
(3) log%28%28a%29%29%2Blog%28%28b%29%29=log%28%28ab%29%29

Use (1) to write 2log%28%28y%29%29 as log%28%28y%5E2%29%29

log%28%28root%283%2Cx%29%29%29-log%28%28y%5E3%29%29%2Blog%28%28y%5E2%29%29

Then use (2) and (3). The "- log" becomes a division and the "+ log" becomes a multiplication:

log%28%28%28root%283%2Cx%29%29%2Fy%5E3%2Ay%5E2%29%29

Then simplify to get the final result.

log%28%28%28root%283%2Cx%29%29%2Fy%29%29