#1**+2 **

(Assuming that log means natural log)

y = 5^{log x}

Take the log of both sides

log y = log( 5^{log x} )

Now we can apply the rule that says: log(x^{y}) = 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...

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

hectictar Mar 22, 2021

#2**+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

kes1968 Mar 22, 2021

#3**+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

hectictar Mar 22, 2021

#4**+2 **

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

kes1968 Mar 22, 2021

#5**+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: \(x=y^\frac{1}{\log 5}\)

hectictar Mar 22, 2021

#6**+3 **

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

kes1968 Mar 22, 2021

#7**+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

hectictar Mar 22, 2021

#8**+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

kes1968 Mar 22, 2021

#9**+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.

Probolobo Mar 22, 2021

#10**+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....

hectictar Mar 22, 2021

#14**+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

Melody Mar 22, 2021