SOLUTION: log6 8+2 log6 3+ log6 12-log6 29

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Question 1210379: log6 8+2 log6 3+ log6 12-log6 29
Found 3 solutions by mccravyedwin, Edwin McCravy, greenestamps:
Answer by mccravyedwin(406) About Me  (Show Source):
You can put this solution on YOUR website!

log%286%2C%288%29%29%22%22%2B%22%222%2Alog%286%2C%283%29%29%22%22%2B%22%22log%286%2C%2812%29%29%22%22-%22%22log%286%2C%2829%29%29

Use log rule:  log%28b%2C%28u%5En%29%29%22%22=%22%22n%2Alog%28b%2C%28u%29%29


log%286%2C%288%29%29%22%22%2B%22%22log%286%2C%283%5E2%29%29%22%22%2B%22%22log%286%2C%2812%29%29%22%22-%22%22log%286%2C%2829%29%29



log%286%2C%288%29%29%22%22%2B%22%222%2Alog%286%2C%283%29%29%22%22%2B%22%22log%286%2C%2812%29%29%22%22-%22%22log%286%2C%2829%29%29

Use log rule:  log%28b%2C%28u%5En%29%29%22%22=%22%22n%2Alog%28b%2C%28u%29%29

log%286%2C%288%29%29%22%22%2B%22%22log%286%2C%283%5E2%29%29%22%22%2B%22%22log%286%2C%2812%29%29%22%22-%22%22log%286%2C%2829%29%29

log%286%2C%288%29%29%22%22%2B%22%22log%286%2C%289%29%29%22%22%2B%22%22log%286%2C%2812%29%29%22%22-%22%22log%286%2C%2829%29%29

Use log rule on first two terms: log%28b%2C%28u%29%29%22%22%2B%22%22log%28b%2C%28v%29%29%22%22=%22%22log%28b%2C%28u%2Av%29%29

log%286%2C%288%2A9%29%29%22%22%2B%22%22log%286%2C%2812%29%29%22%22-%22%22log%286%2C%2829%29%29

Use it again on the first two terms:

log%286%2C%288%2A9%2A12%29%29%22%22-%22%22log%286%2C%2829%29%29

Use log rule: log%28b%2C%28u%29%29%22%22-%22%22log%28b%2C%28v%29%29%22%22=%22%22log%28b%2C%28u%2Fv%29%29

log%286%2C%28%288%2A9%2A12%29%2F29%29%29

Write 8=2%5E3, 9=3%5E2, 12=4%2A3=2%5E2%2A3

Then 8%2A9%2A12%22%22=%22%22%282%5E3%29%2A%283%5E2%29%2A%282%5E2%2A3%29%22%22=%22%222%5E5%2A3%5E3%22%22=%22%222%5E2%2A2%5E3%2A3%5E3+=+%282%5E2%2A%282%5E3%2A3%5E3%29%29=4%2A%282%2A3%29%5E3+=+4%2A6%5E3

So

log%286%2C%28++%288%2A9%2A12%29%2F29++%29%29%22%22=%22%22log%286%2C%28%284%2A6%5E3%29%2F29%29%29%22%22=%22%22
log%286%2C%28++%284%2F29%29%2A6%5E3+%29%29%22%22=%22%22log%286%2C%284%2F29%29%29%22%22%2B%22%22log%286%2C%286%5E3%29%29%22%22=%22%22log%286%2C%284%2F29%29%29%22%22%2B%22%223%2Alog%286%2C%286%29%29%22%22=%22%22log%286%2C%284%2F29%29%29%22%22%2B%22%223%2A1%22%22=%22%22log%286%2C%284%2F29%29%29%22%22%2B%22%223

Approximately 1.894382029

Edwin

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
I was wondering about this problem since it didn't simplify into anything nice.

Maybe all your teacher wanted was only for you to get the decimal approximation:





log%286%2C%28+%288%2A3%5E2%2A12%29%2F29+%29%29%22%22=%22%22log%286%2C%28864%2F29%29%29

Then use change of base formula, change to natural logs:

log%286%2C%28864%2F29%29%29%22%22=%22%22ln%28864%2F29%29%2Fln%286%29%22%22=%22%221.894382029

Edwin

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Perhaps your handwriting is as sloppy as mine and the "29" is supposed to be "24". Then the expression simplifies nicely.

log6 8+2 log6 3+ log6 12-log6 24
log%286%2C8%29%2B2log%286%2C3%29%2Blog%286%2C12%29-log%286%2C24%29
log%286%2C%28%28%288%29%283%5E2%29%2812%29%29%2F24%29%29
log%286%2C%28864%2F24%29%29
log%286%2C36%29
2