# What is the inverse of y=3log(5x)+x^3? ?

Oct 1, 2017

$x = 3 \log \left(5 y\right) + {y}^{3}$

#### Explanation:

Given:

$y = 3 \log \left(5 x\right) + {x}^{3}$

Note that this is only defined as a real valued function for $x > 0$.

Then it is continuous and strictly monotonically increasing.

The graph looks like this:
graph{y = 3log(5x)+x^3 [-10, 10, -5, 5]}

Therefore it does have an inverse function, whose graph is formed by reflecting about the $y = x$ line...
graph{x = 3log(5y)+y^3 [-10, 10, -5, 5]}

This function is expressible by taking our original equation and swapping $x$ and $y$ to get:

$x = 3 \log \left(5 y\right) + {y}^{3}$

If this were a simpler function then we would typically want to get this into the form $y = \ldots$, but that is not possible with the given function using standard functions.