# How would you graph y= -lnx ?

Jul 28, 2017

Take the graph of $y = {e}^{x}$, reflect it in the line $y = x$, then in the $x$ axis.

#### Explanation:

Do you know what the graph of $y = {e}^{x}$ looks like?

graph{y=e^x [-10, 10, -5, 5]}

• It is monotonically increasing.
• It is always greater than $0$, so lies completely above the $x$ axis.
• It is rapidly asymptotic to the $x$ axis for negative values of $x$.
• It intersects the $y$ axis at $\left(0 , 1\right)$.
• It grows very rapidly for positive values of $x$.

Next note that $\ln x$ is the inverse function of ${e}^{x}$.

So the graph of $y = \ln x$ can be found by swapping $x$ and $y$, that is by reflecting the above graph in the diagonal line $y = x$, to get:

graph{y=ln x [-10, 10, -5, 5]}

Note that:

• It is monotonically increasing.
• It is only defined for $x > 0$, so the graph lies entirely to the right of the $y$ axis.
• It has a vertical asymptote at $x = 0$.
• It intersects the $x$ axis at $\left(1 , 0\right)$.
• It grows very slowly for positive values of $x$.

Finally, to get the graph of $y = - \ln x$ we just have to reflect the above graph in the $x$ axis to get:

graph{y=-ln x [-10, 10, -5, 5]}

Note that:

• It is monotonically decreasing.
• It is only defined for $x > 0$, so the graph lies entirely to the right of the $y$ axis.
• It has a vertical asymptote at $x = 0$.
• It intersects the $x$ axis at $\left(1 , 0\right)$.
• It grows more negative very slowly for positive values of $x$.