Question

if 3^a = 4^b

Find the value of this amount:-

9^a/b - 16^b/a

Answer 1

@ShadowLegendX

Answer 2

in such questions using LN helps so much.
Are you ready to start thinking with me?

Answer 3

ok i'm ready

Answer 4

well, do you know what is LN?
Also tell me what is the other form of the phrase below:
\[\ln a^{n}=?\]

Answer 5

i don't know what is LN

Answer 6

logarithm

Answer 7

ok i know log but i don't know what is in ??

Answer 8

what is the differance between log and in ?

Answer 9

it is the Natural Logarithm.
ok, do you know the Logarithm formulas?
for example, can you write the phrase below in a simpler way:
\[loga ^{N}=??\]

Answer 10

i don't know
can you help me

Answer 11

OK,
\[loga ^{N} = N.loga\]

You got it?

Answer 12

yes

Answer 13

well!
You have this equation:
\[3^{a}=4^{b}\]
Apply Log to both sides. What will be the result?

Answer 14

by the sentence "Apply Log to both sides" I mean simply add the sign of logarithm to both sides in the equation above.

Answer 15

ok . continue plz

Answer 16

a. log 3 = b.log 4

Answer 17

is that correct.?

Answer 18

yes, that's right,

now solve for a/b

Answer 19

i don't know how to solve for a/b

Answer 20

divide both sides by log 3,
then divide both sides by b

Answer 21

how???
can you sshow me how

Answer 22

a. log 3 = b.log 4

(dividing both sides of the equation by log 3)

a. log 3 / log 3 = b.log 4 / log 3

(simplifying the lefthand side)

a = b. log 4 / log 3

now divide by b

Answer 23

@A.ahmed

Answer 24

great i'm sorry the connection went out

Answer 25

you are amazing

Answer 26

EXCUSE ME, A.Ahmed!
Can you wait a minute so me and other viewers here talk about it?

Answer 27

I guess there's a simpler way for solving it without using Logarithms. Please correct it if you see I'm making a mistake.

Answer 28

like this...

\[3^a = 4^b\]

square both sides

\[(3^a)^2=(4^b)^2\]

\[3^{2a}=4^{2b}\]

\[(3^2)^a=(4^2)^b\]

\[(9)^a=(16)^b\]

raise to the \(1/b\) power

\[(9)^{a/b}=16\]

raise to the \(1/a\) power

\[9=(16)^{b/a}\]


thus

\[9^{a/b}-16^{b/a}=16-9=7\]

Answer 29

\[3^{a}=4^{b} ==> 9^{a}=16^{b}\]

Answer 30

YES! Exactly like what Zarkon said, without using Logarithm

Answer 31

mr. yavar .i want to know the answer using logarithm .this answer is more diffucalr

Answer 32

mr. @UnkleRhaukus can you continue the answer plz

Answer 33

You're correct. Zarkon has given the simplest way to solve this. Using logarithm and its rules will make it more difficult to solve it.

Answer 34

If you can understand it completely we can close this question, if you need complementary explanations on his solution, i can help you.

Answer 35

mr. yavar can you continue the answer using logarithm by divide by b

Answer 36

actually in the middle of solving it, the same solution came to my mind and i didn't continue it.
Do you insist to know a solution using logarithms?

Answer 37

yes

Answer 38

Ok,
so:
\[a \log3 = b \log4\] SO \[\frac{ a }{ b }=\log_{3} 4\]
AND\[\frac{ b }{ a }=\log_{4} 3\]
now we can substitute these equations in the phrase below:
9^a/b - 16^b/a --> 3^2a/b-4^2b/a
which after substitution becomes the attached file here.
It doesn't show the equations well when I use it. So I have attached the file below:

Answer 39

I have written the the most important formula I used, but I guess as long as you're not comfortable with logarithm rules, it's better to do it like Zarkon which is very much simpler.

Do Great!

Answer 40

you are great @Yavar . thank you too much

Answer 41

:) you're welcome.