How do you sketch the graph of $y = {log}_{\frac{1}{2}} x$ $y=log_(1/2)x$ and $y = {(\frac{1}{2})}^{x}$ $y=(1/2)^x$ ?

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Answer 1 · 2016-10-29T07:57:11

See the explanation and the Socratic graphs.

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

two are one and the same...

But the first is got from the second by the swapping

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

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 $- or + 90^{o}$ $-or+90^o$ , about the origin.

Graph of $y = {log}_{\frac{1}{2}} x$ $y = log_(1/2)x$ :

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

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

$(x, y) :$ $(x, y):$

$(-oo, oo)...(.-5, 32) (-4, 16) (-8, 3) (-4, 2), (-1, 2) (0, 1)$

$(1, 1/2) (2, 1/4) (3, 1/8) (4, 1/16) (5, 1/32) ...(oo, 0)$

Graph of $y = (1/2)^x$ :

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

How do you sketch the graph of y=log_(1/2)xy=log12xy=log_(1/2)x and y=(1/2)^xy=(12)xy=(1/2)^x?