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+1
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Guys please solve this question and let me know the correct option

please its a request here the link of question if you cant see it (https://ibb.co/ZWftpcR)

Mar 22, 2021
edited by kes1968  Mar 22, 2021

#1
+9279
+2

(Assuming that  log  means natural log)

y   =   5log x

Take the log of both sides

log y   =   log( 5log x )

Now we can apply the rule that says:  log(xy)  =  y log x

log y   =   log x log 5

Divide both sides by  log 5

log y / log 5   =   log x

Do the reverse of natural log to both sides (make both sides the exponent of e )

e^( log y / log 5  )   =   e^log x

Now the right side simplifies to just  x

e^( log y / log 5  )   =   x

So we have...

$$x\ =\ e^{\frac{\log y}{\log 5}}\\~\\ x\ =\ (e^{\log y})^{\frac{1}{\log 5}}\\~\\ x\ =\ y^{\frac{1}{\log 5}}$$

Mar 22, 2021
#2
+365
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brother what you have found is the same function just in terms y , but inverse function is symmetrical about y=x axis ,

also f(f^-1(x))=x which is not satisfied ,

by the way if i do the same operation and interchange x and y i get 2 and 4 option , that satifies both conditions

Mar 22, 2021
#3
+9279
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The inverse function can be graphed by:    $$y=x^{\frac{1}{\log5}}$$

Here's a graph:

https://www.desmos.com/calculator/aijpzpouk1

In other words...

If   $$f(x)=5^{\log x}$$    then the inverse is   $$f^{-1}(x)=x^{\frac{1}{\log 5}}$$

And...

$$f(f^{-1}(x))\ =\ f(x^{\frac{1}{\log 5}})\ =\ 5^{\log(x^{\frac{1}{\log 5}})}\ =\ 5^{\frac{\log x}{\log 5}}\ =\ 5^{\log_5 x}\ =\ x$$

The options leave it in the form that is solved for  x, and so I left it like that to match the options

Mar 22, 2021
#4
+365
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brother so what you feel (as a math expert) should be the right answer to the given question ,

Mar 22, 2021
#5
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I'm not sure what you meant by "by the way if i do the same operation and interchange x and y i get 2 and 4 option , that satifies both conditions"....but if this didn't answer your question then please feel free to ask for more clarification!

And I think the answer is option 1:  $$x=y^\frac{1}{\log 5}$$

Mar 22, 2021
#6
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brother , if you see your second answer and change it in terms of x , wont you get option 2 and option 4 ,

Mar 22, 2021
#7
+9279
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Hmm...actually....I see what you mean.... (maybe I made the question harder than it has to be!)

I take back my original answer!! Now I think the answer is option 4

Mar 22, 2021
#8
+365
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brother do you really believe its option 4 or just to keep my heart , please clarify if you still believe the answer is 1 , if yes then please prove the same to me as well

Mar 22, 2021
#9
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For computing the inverse function, the plan I know is
1. interchange x & y

2. solve for y

but actually, after 1. you already have a term for the inverse function. It's just not written in the "usual" way, wich is y=f(x).

Answer for is exactly what you get after interchanging x&y, so the correct answer is answer 4.

Mar 22, 2021
#10
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I do agree Probolobo, but then the confusing thing is that

$$5^{\log y}\ =\ y^{\log 5}$$

Which means option 2 and option 4 are the same function and so are equivalent....

Mar 22, 2021
#12
+1740
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That's correct, then there are actually 2 correct answers, 2 & 4. Didn't see that equivalence on first glance ;)

Probolobo  Mar 22, 2021
#11
+365
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so whats the final answer so that i can challenge the answer key?

Mar 22, 2021
#13
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If I had to guess right now, I would guess option 4. My next guess would be option 1.

But I am honestly not sure!! I think this is a bad question.

Can you let us know when you find out the "correct" answer according to the answer key?

hectictar  Mar 22, 2021
#14
+112907
+3

I think you are making hard work of it

the inverse of

$$y=5^{logx}$$

is simply

$$x=5^{logy}$$

You just have to switch the x and y over.

there would be restrictions on x and on y but the question isn't worrying about that.

Here is the graphs

https://www.desmos.com/calculator/4ftfa2bny7

See they are reflections of each other about y=x

Mar 22, 2021
#15
+365
+2

another question , another superb explanation from mod

Melody you are simply awesome ! thanks for the help mate

kes1968  Mar 22, 2021
edited by kes1968  Mar 22, 2021
#16
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You are welcome Kes

Melody  Mar 22, 2021