+0  
 
+1
104
16
avatar+365 

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
avatar+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
avatar+365 
+2

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
avatar+9279 
+3

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 smiley

 Mar 22, 2021
 #4
avatar+365 
+2

brother so what you feel (as a math expert) should be the right answer to the given question ,

 Mar 22, 2021
 #5
avatar+9279 
+1

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
avatar+365 
+3

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
avatar+9279 
+1

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
avatar+365 
+1

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
avatar+1740 
+1

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
avatar+9279 
+2

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
avatar+1740 
+1

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
avatar+365 
+1

so whats the final answer so that i can challenge the answer key?

 Mar 22, 2021
 #13
avatar+9279 
+1

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
avatar+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
avatar+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
avatar+112907 
+1

You are welcome Kes

Melody  Mar 22, 2021

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