Graph exponential function

Question

Hi peeps, can somebody please help me with this exercise? I'm having a hard time with it and am going nowhere.

Thank you in advance!!

Graph the function  and its reflection about the line y=x on the same axis, and give the x-intercept of the reflection. Prove that . 

Answer 1

Hi Joshua C.,

If you're like most people, you'd prefer to think about numbers greater than 1 being exponentiated? So here, the function is = 5* (2)^x . To get the reflection, substitute y for x and x for y. But that's not standard form; so you could express taking the ln of each side. The x-intercept of the reflection = the y-intercept of the original function = 5*2^0 = 5, since anything ^0 = 1 (except 0, that's undefined).

So using standard math, the proof for the equation is simple: take the ln of each side, and simplify. Any thing in the exponent pops directly out as a factor; anything NOT in the exponent stays under the ln. So ln(a^x) = x* ln(a), directly. That's because taking the ln of something is the inverse operation of taking the exp of it. You just have to be careful in distinguishing what is the "something"!

-- Cheers, -- Mr. d.

Answer 2

I just answered this question which has been submitted twice.

Answer 3

The graphing you will have to do for yourself (try DESMOS), but I can help you with it.

The reflection of a function across the x-axis is the graph of the inverse function.

To get the inverse function interchange the roles of x and y in the original function and solve for y. In this case

x=5(.5)^y => ln x = ln 5 + y ln .5

y=(ln x - ln 5)/ln .5

When you plot thse 2 functions on the same axis you will see that they are symmetric with respect to the x axis.

The proof you require; take the ln of a^X and then raise e to that ln and remember the definition of e^x and ln x.