Question

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.

log_3(567)-log_3(7)

Answer 1

you could use log(a) - log(b) = log(a/b)

Answer 2

as a start

Answer 3

http://www.chilimath.com/algebra/advanced/log/images/rules%20of%20exponents.gif <--- notice the 2nd rule listed there

Answer 4

log_3(567^3/7^3) ?

Answer 5

no ^3 just 567/7

Answer 6

Apparently that answer is missing inputs

Answer 7

I'm not sure what that even means

Answer 8

we are not done yet

Answer 9

(this homework is online)

Answer 10

log_3(567)-log_3(7) = ?

Answer 11

use the rule log(a) - log(b)= log(a/b)

Answer 12

right so what would you do from log_3(81)? 81^X=3?

Answer 13

or wait 3^x=81? then solve?

Answer 14

yes, you get log_3(81)

if you can write 81 as 3^x (and we can) we can simplify some more

Answer 15

so is the answer what x is, which is 4?

Answer 16

we replace 81 with 3^4 in log_3(81)

log_3(3^4)

now use the rule log(a^b) = b log(a)

Answer 17

in other words
log_3(81)= 4 log_3(3)

last rule (finally!) log_3(3) = 1

Answer 18

4 ended up being the correct answer

Answer 19

yes
4 log_3(3) = 4*1 = 4

so we found
log_3(567)-log_3(7) = 4

Answer 20

oh okay thats where the one comes from I get it now!

Answer 21

That one makes sense to me. How about this one?

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.

8ln(x+14)-5ln(x)

Answer 22

you can "bring in" the 8 (and the 5) using
a log(b)= log(b^a)

Answer 23

so log(x+14)^8?

Answer 24

yes, but it's more clear to write it
log( (x+14)^8 )
because the 8 is inside the log

Answer 25

do the same for 5 log(x)

Answer 26

okay so I have log((x+14)^8)-log((x^5)). now what?

Answer 27

now use log(a) - log(b) = log(a/b)

and we should use ln (which is short of log base e) to match the problem

Answer 28

Can you show me how to do that with this one I'm confused lol

Answer 29

log( (hairy expression)) - log((messy stuff)) = log( (hairy expression)/(messy stuff) )

Answer 30

log((x+14)^8/(x^5)) ?

Answer 31

yes. Just to be clear, it is
\[\ln \left( \frac{ (x+14)^8 }{ x^5 } \right)\]

Answer 32

Thank you. I missed class so I am unaware of all the properties. Can you please help with one more:)

Answer 33

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.

Answer 34

I attatched it as a file

Answer 35

I am suppose to fill in the blank after log_7

Answer 36

do it step by step
first, "condense" log_7(x) - log_7(y)

Answer 37

log_7(x/y)

Answer 38

now we have
(1/7) log_7(x/y)

use a log(b) = log(b^a)

in this case, b is (x/y)

Answer 39

log((x/7)^1/7)

Answer 40

x/y***

Answer 41

yes , and base 7 i.e. log_7

Answer 42

\[\log_7\left( \left( \frac{ x }{ y } \right)^{\frac{ 1 }{ 7 }} \right)+5 \log_7\left( x-6\right)\]

Answer 43

now condense the 2nd term

Answer 44

using a log(b) = log(b^a)

Answer 45

log_7((x-6)^5)

Answer 46

ok, last step
log_7((x/y)^1/7) + log_7( (x-6)^5)

use log(a)+log(b) = log(a b) (i.e multiply a times b)

Answer 47

log_7((x/y)^1/7 * (x-6)^5)

Answer 48

or is there a way to actually simplify that

Answer 49

no. that is what it is. A trifle ugly but we mustn't be rude to its face.

Answer 50

haha okay so that's the answer?

Answer 51

yes, that is the answer.

if you want more background on this, you can watch
http://www.khanacademy.org/math/algebra2/logarithms-tutorial/logarithm_properties/v/introduction-to-logarithm-properties

Answer 52

thank you very much. You have been a huge help. I have more questions but I must be waring you out, should I repost?