Question

Suppose 0 < A < AB < 1 < B
Rank the following five expressions from least to greatest by selecting the correct order of the expressions in the inequality.
1) 0
2) Log(A)
2) Log(b)
3) Log(A)+Log(B)
5) Log(B^A)

Answer 1

I dont see any difference between 3rd and 4th expressions ?

Answer 2

Sorry, my bad! Does that make more sense?

Answer 3

Since A and AB are less than 1, log(A) and log(AB) will be negative

Answer 4

see if u can sort them ..

Answer 5

What are the rules for determining positive/negative based on <=>1?

Answer 6

Like I get that a and a*b are gunna be fractions but I dont understand once their put into a log how you can tell what they will come out to be?

Answer 7

say \(a \lt 1\), consider below :

\(\log_{10} (a)\)

Answer 8

look at the graph, whats the value of `log` function when x is less than 1 ?

Answer 9

if u dont like graphs :

\(\log_{10} (a) = x \implies a = 10^x\)

since \(a\) is a fraction, \(x\) needs to be a negative (why ?)

Answer 10

\(1 < B^{A} < B\)

This makes: \(0 < A < AB < 1 < B^{A}< B \)

The logarithm function preserves order. This is a property that makes it such a valuable transformation in many fields and exercises. Thus:

\(log(A) < log(AB) < log(1) < log(B^{A})< log(B)\) -- We have to lose the zero (0).

A little logarithm magic, gives:

\(log(A) < log(AB) = log(A) + log(B) < log(1) = 0 < log(B^{A}) = A\cdot log(B)< log(B)\)

Note: It is generally considered bad form to change variables in the middle of a problem. Don't start talking about "B" and then end up talking about "b".

Answer 11

......

Answer 12

-_-_-

Answer 13

ask a question if u dont understand lol.. hard to understand dot language :P

Answer 14

That's because I have no idea what to do with the information i was just spewed at lol...I think we should just redirect back to your version lol

Answer 15

There is not much there.

1) Put \(B^{A}\) in the list.
2) Apply the logarithm function.
3) Use a few logarithm rules.

Look at each piece. Let it soak in.

Answer 16

first things first : you need to know when a log value is negative, and when it is positive

Answer 17

did u get when it would be negative/positive ?

Answer 18

I understand in exponential form but I guess I'm just having difficulty applying that to log form

Answer 19

okay, the graph was not useful is it ?

Answer 20

no not really, we didn't spend much time on graphing

Answer 21

okay, then discard the graph

lets see how to figure out the log value from exponent perspective

Answer 22

\(\large \log_{10} (a) = x ~~~~ \implies~~~~ a = 10^x\)

Answer 23

fine with this right ?

Answer 24

yeah, and since a is a fraction that would make x negative eg Log(a)=-# correct?

Answer 25

you got it !

Answer 26

\(a = 10^x\)

for example, when \(x = -1\) :

\(a = 10^{-1} = 0.1\)

which is a fraction !

Answer 27

so exponent has to be negative, for the value to evaluate to a fraction

Answer 28

log value = exponent = x