When cell phones first began to be used, news announcers excitedly reported that the use of cell phones were growing exponentially.

Question

Did he really understand what that means? Was he correct and how do you know?

 

Also, how are exponential and logarithmic functions alike and different? 

Answer 1

The news announcers did know what they were talking about when the use of cell phones grew exponentially, and they were correct. It is just like any new invention that gets introduced into the market place. The use grows exponentially if the invention solves a problem and becomes high demand. Like with the cell phone, it made it more convenient for people to contact each other on more consistent basis rather than relying on what you call the land-line. We didn't call them land-lines. We just call them telephones in which you had to plug them into a phone line within your house. The invention of answering machines did away with missed calls and enabled people to get back with the caller at a later time. 

 

Now, to answer how exponential and logarithmic functions alike and different. The way they are alike is that they are both exponential functions. The logarithmic function is the inverse of an exponential function. Here's an example,

 

f(x)=a^x   and g(x)=log_ax   f(g(x))=a^log_ax = x and g(f(x))=log_aa^x=x*log_aa =x*1=x

 

The way to prove the first identity is true is let y = f(g(x)) = a^log_ax.

 

Take log_a of both sides.

 

log_ay=log_aa^(log_ax)=log_ax*log_aa=log_ax*1=log_ax This implies log_af(g(x))=log_ax Thus, f(g(x))=x.

 

Therefore, exponential and logarithmic functions are inverse functions to each other. The differences are they are inverse functions.