SOLUTION: Hello! A function f(x) is given. f(x) = log5(log10(x)) (a) Find the domain of the function f (b) Find the inverse function of f. f -1(x) = thanks for your help!

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Question 198808: Hello!
A function f(x) is given.
f(x) = log5(log10(x))
(a) Find the domain of the function f
(b) Find the inverse function of f.
f -1(x) =
thanks for your help!

Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
a)

Since you cannot take the log of 0 or a negative value, this means that log%2810%2C%28x%29%29%3E0 and x%3E0 (for the outer and inner logs)


If we solve log%2810%2C%28x%29%29%3E0, we get x%3E1. So combining the inequalities x%3E0 and x%3E1 (ie perform a set union), we get x%3E1.

So the domain of f%28x%29=log%285%2C%28log%2810%2C%28x%29%29%29%29 is x%3E1


------------------

b)

f%28x%29=log%285%2C%28log%2810%2C%28x%29%29%29%29 Start with the given function.


y=log%285%2C%28log%2810%2C%28x%29%29%29%29 Replace f(x) with "y"


x=log%285%2C%28log%2810%2C%28y%29%29%29%29 Switch each x and y


5%5Ex=log%2810%2C%28y%29%29 Rewrite the equation using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x


10%5E%285%5Ex%29=y Rewrite the equation again using the property: log%28b%2C%28x%29%29=y ====> b%5Ey=x


So the inverse function is