What is the form of the inverse of the function y = k^x?

To find the inverse of the function ( y = k^x ), we start by switching the roles of ( x ) and ( y ). This means that we will solve for ( y ) in terms of ( x ).

The equation can be rewritten as ( x = k^y ). To solve for ( y ), we want to express ( y ) in logarithmic form. By taking the logarithm base ( k ) of both sides, we get:

[ \log_k(x) = y ]

This means that the inverse function is expressed as ( y = \log_k(x) ).

Now, looking closely at the choices presented, the correct form of the inverse is ( y = \log_k(x) ), as it represents the relationship between ( x ) and ( y ) after interchanging them and solving for ( y ).

The other expressions given—such as ( y = k^{-x} ) or ( y = -k^x )—do not represent the inverse of the original exponential function, as they do not correctly account for the relationship established by taking logarithms.

Thus, the inverse of ( y = k^x