(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...
\(x\ =\ e^{\frac{\log y}{\log 5}}\\~\\ x\ =\ (e^{\log y})^{\frac{1}{\log 5}}\\~\\ x\ =\ y^{\frac{1}{\log 5}}\)
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
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
brother so what you feel (as a math expert) should be the right answer to the given question ,
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: \(x=y^\frac{1}{\log 5}\)
brother , if you see your second answer and change it in terms of x , wont you get option 2 and option 4 ,
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
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
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.
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....
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