# How do you sketch the graph of y=log_(1/2)x and y=(1/2)^x?

Oct 29, 2016

See the explanation and the Socratic graphs.

#### Explanation:

The inverse of the second is $x = {\log}_{\frac{1}{2}} y$ and the graphs of these

two are one and the same...

But the first is got from the second by the swapping

$\left(x , y\right) \to \left(y , x\right)$..

Conventionally ( traditionally ), many call each of the given relations

as the inverse relation for the other.

Separately, each graph can be obtained from the other by rotation

through $- \mathmr{and} + {90}^{o}$, about the origin.

Graph of $y = {\log}_{\frac{1}{2}} x$:

x = 0 ( y-axis) is asymptotic and $x > 0$...
graph{y+1.44 ln (x)=0[0 20 -5 10]}

A short Table for the second $y = {\left(\frac{1}{2}\right)}^{x}$ is

$\left(x , y\right) :$

$\left(- \infty , \infty\right) \ldots \left(. - 5 , 32\right) \left(- 4 , 16\right) \left(- 8 , 3\right) \left(- 4 , 2\right) , \left(- 1 , 2\right) \left(0 , 1\right)$

$\left(1 , \frac{1}{2}\right) \left(2 , \frac{1}{4}\right) \left(3 , \frac{1}{8}\right) \left(4 , \frac{1}{16}\right) \left(5 , \frac{1}{32}\right) \ldots \left(\infty , 0\right)$

Graph of $y = {\left(\frac{1}{2}\right)}^{x}$:

y = 0 ( x-axis ) is asymptotic and $x > 0$..
graph{ y-(0.5)^x=0[-5 25 -10 10]}