Question

Expand completely: log

Answer 1

\[\log _{3}(x^{2}y^{3}) \over (_{z^4})\]

Answer 2

@jim_thompson5910 I don't have a clue

Answer 3

is everything in the log? or is it just the expression x^2y^3 only in the log?

Answer 4

everything is in the log

Answer 5

I couldn't figure out how to get the z^4 in the prenthesis

Answer 6

So the full problem is actually

\[\Large \log_{3}\left(\frac{x^{2}y^{3}}{z^{4}} \right )\]

Answer 7

yes

Answer 8

I am having a hard time with these

Answer 9

first use the idea that log(A/B) = log(A) - log(B)


\[\Large \log_{3}\left(\frac{x^{2}y^{3}}{z^{4}} \right )\]


\[\Large \log_{3}\left(x^{2}y^{3}\right )-\log_{3}\left(z^{4} \right )\]

Answer 10

Then use the idea that log(AB) = log(A) + log(B)


\[\Large \log_{3}\left(\frac{x^{2}y^{3}}{z^{4}} \right )\]


\[\Large \log_{3}\left(x^{2}y^{3}\right )-\log_{3}\left(z^{4} \right )\]


\[\Large \log_{3}\left(x^{2}\right )+\log_{3}\left(y^{3}\right )-\log_{3}\left(z^{4} \right )\]

Answer 11

ok let me see from here

Answer 12

why at one point we were subtracting and then it was switched to addition

Answer 13

I expanded

\[\Large \log_{3}\left(x^{2}y^{3}\right )\]

into

\[\Large \log_{3}\left(x^{2}\right )+\log_{3}\left(y^{3}\right )\]

Answer 14

So that's where the addition came from (the subtraction is still there)

Answer 15

I'm just ignoring it for now

Answer 16

what happened to z^4

Answer 17

like I said, I'm ignoring it for now and only focusing on expanding \[\Large \log_{3}\left(x^{2}y^{3}\right )\]

Answer 18

to show where the addition came from

Answer 19

ok

Answer 20

so where do I go from here

Answer 21

ok let me retype all of the steps so far

Answer 22

\[\Large \log_{3}\left(\frac{x^{2}y^{3}}{z^{4}} \right )\]


\[\Large \log_{3}\left(x^{2}y^{3}\right )-\log_{3}\left(z^{4} \right )\]


\[\Large \log_{3}\left(x^{2}\right )+\log_{3}\left(y^{3}\right )-\log_{3}\left(z^{4} \right )\]

Answer 23

The last step is to pull down the exponents using the idea that log(x^y) = y*log(x)

Answer 24

So the full step by step answer is

Answer 25

\[\Large \log_{3}\left(\frac{x^{2}y^{3}}{z^{4}} \right )\]


\[\Large \log_{3}\left(x^{2}y^{3}\right )-\log_{3}\left(z^{4} \right )\]


\[\Large \log_{3}\left(x^{2}\right )+\log_{3}\left(y^{3}\right )-\log_{3}\left(z^{4} \right )\]


\[\Large 2\log_{3}\left(x\right )+3\log_{3}\left(y\right )-4\log_{3}\left(z \right )\]

Answer 26

Which means that

\[\Large \log_{3}\left(\frac{x^{2}y^{3}}{z^{4}} \right )\]

fully expands out to

\[\Large 2\log_{3}\left(x\right )+3\log_{3}\left(y\right )-4\log_{3}\left(z \right )\]

Answer 27

ok thank you @jim_thompson5910

Answer 28

yw