exponential decay and logarithm

Question

The inverse of exponential growth function y=a^x is the logarithmic function y=log_a(x). But is there an inverse for a decay exponential function y=a^(-x)? and how its graph would be?

Answer 1

You can figure that out by switching your x's and y's as you would any other inverse, with your log insights.

 

• y=a^-x
• x=a^-y (flip x's and y's)
• log_ax = -y (logarithmic equivalent)
• y=-log_ax

 

This would graph as the negative version of your log function, so basically just flipped (reflected) over the x axis

Answer 2

a^-x = ((a)^-1)^x = (1/a)^x , so the inverse function for y = a^-x is

y = log_(1/a)x.  The graph of y = log_(1/a)x is the reflection of the graph of y = (1/a)^x through the line y = x.