Question

how do I write 4 logb^m + 1/2logb^n - 3logb^2p as one log?

Answer 1

maybe if I draw this it will make more sense

Answer 2

is the base to the log is b or is it b^m ( b to the power m)

Answer 3

here let me just take a picture of the problem

Answer 4

there

Answer 5

Is that ok @rajee_sam ?

Answer 6

Ok, hang on for the ride

Answer 7

\[4\log_{b} m + \frac{ 1 }{ 2 }\log_{b} n - 3 \log_{b} 2p\]

Answer 8

oh dear..

Answer 9

do you know the basic log rules

Log A + Log B = Log AB
Log A - Log B = Log (A/B)
m Log A = Log A^m

Answer 10

all should have the same base

Answer 11

here all logs have the same base which is 'b'

Answer 12

ok

Answer 13

I'm following you so far.

Answer 14

now lets apply the rules one by one.

first we need to simplify each log separately before we can combine them.

For convenience I am not writing the b. since all bases are same I will write it at the end. But in your work you write it when you are solving in each and every step.

Now,

How can you rewrite 4 log m

Answer 15

which rule can we apply?

Answer 16

The first one?

Answer 17

First one has two logs added. here we have only one ---> m Log A

Answer 18

how can I rewrite this single log?

Answer 19

ok, and you just moved the 4 to be the exponent of m?

Answer 20

\[4 \log_{b} m = \log_{b} m ^{4}\]

Answer 21

ok

Answer 22

yeah similarly how can I rewrite the other two independently

Answer 23

\[\frac{ 1 }{ 2 }\log_{b} n = ?\]

Answer 24

logb n^1/2 ?

Answer 25

Sorry I don't know how to use the tools so well

Answer 26

yes and the third one

Answer 27

That's alright

Answer 28

logb 2p^-3?

Answer 29

\[\log_{b} m ^{4} + \log_{b} n ^{\frac{ 1 }{ 2 }} - \log_{b} (2p)^{3}\]

Answer 30

I will keep the negative sign and just take the 3 to the power

Answer 31

ok, fabulous. But I'm not done, right? I still need to write it as one log.

Answer 32

not yet

Answer 33

now we have to combine them two at a time

Answer 34

once you are familiar with it you can do all at one go. But to understand better we will do two at a time now

Answer 35

ok

Answer 36

Now to combine 1st and 2nd

we have something like Log A + Log B , so using the rule how can write it as a single log?

Answer 37

Log A + Log B = Log AB

Answer 38

they are both log b, so that stays the same, right? And then do I just put m^4 and n^1/2 next to eachother?

Answer 39

like logb m^4 n^1/2?

Answer 40

yes. you are writing them together with a multiplication sign in between them

Log A + Log B = Log AB ; AB actually means A x B

Answer 41

Yes you are right

Answer 42

\[\log_{b} m ^{4}n ^{\frac{ 1 }{ 2 }}\]

Answer 43

YAY

Answer 44

now again you have two logs this one you just wrote and the 3rd one which is of the form Log A - Log B = Log (A/B)

Answer 45

now use the other rule to write them together

Answer 46

ok so logb m^4 n^1/2 / logb (sp)^3

Answer 47

I meant 2 not s

Answer 48

no there should be only one log in your answer

Answer 49

log A - log B = log (A/B)

Answer 50

you have written log A / Log B

Answer 51

\[\log_{b} \frac{ m ^{4} n ^{\frac{ 1 }{ 2 }} }{ (2p)^{3} }\]

Answer 52

and voila that is your answer

Answer 53

ahh ok awesome, thanks so much!!!!