Find the inverse of y=log9x

Question

 The nine is the base, and x is the argument/power.

Answer 1

See https://www.desmos.com/calculator/lndyevexyt for the graph of this function and its inverse.

 

One way to find the inverse of a function is to interchange x and y, then solve for y.

 

So,

 

x=log₉y

 

To solve for y rewrite in exponential form:

 

y = 9^x

 

You have to pay attention to the domain/range of the original function and whether the range of the original does indeed become the domain of the inverse. For example, even though y = x² is a function it is not one to one, e.g. f(-2) = 4 and so does f(2). So the domain of the original has to be restricted in order to generate its inverse. This is sometimes called the horizontal line test, i.e. if a horizontal line passes through the graph in more than one point, the function does not have an inverse over its entire domain.

Answer 2

Interchange  both X and y

 

giving you

 

x=log9y

 

using the rule for logarithms

 

y= 9^x

 

thats your inverse.