Use f(x) = log3(9x) and g(x) = 3^x

Question

Use f(x) = log₃(9x) and g(x) = 3^x

(a) Find f(g(x)) and simplify

For this one I am confused because I am not sure how to deal with logs of a different base but this is my attempt.

log₃(9(3^x))

xlog₃ (27)

but then I'm stuck...

(b) Find the range of y = f(x) + g(x)

on this I would have no idea because I have a log adding a number, so if you could please explain the steps to take on each part I would be super grateful.

Thank you.

Answer 1

f(x) = log₃(9x)   g(x) = 3^x

 

(a). f(g(x)) = f(3^x) = log₃[9(3^x)]

 

                          = log₃[3²3^x]

 

                          = log₃[3^x+2] = x+2   Recall:  log_a(a^r) = r

 

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

(b). f(x) + g(x) = log₃(9x) + 3^x

 

     the range of y = 3^x is all positive reals

 

     the range of log₃(9x) is all reals

 

     the range of f(x) + g(x) is (-∞, ∞)

 

                        

Answer 2

I will help you with part (a):

 

f(x) = log₃(9x)

 

g(x) = 3^x

 

f(g(x)) = log₃[9g(x)] = log₃[9(3^x)] = log₃9 + log₃(3^x) = 2 + x*log₃3 = 2 + x(1) = 2 + x