Question

match each equation with a graph above
$y = \ln{(2 - x)}$
a. green
$y = 1.6\ln{(x)}$
b. blue
$y = \log{(1 - x)}$
c. black

Explanation:

Step1: Analyze $y=\ln(2-x)$

First, find the domain: $2-x>0 \implies x<2$. Find the x-intercept: set $y=0$, $\ln(2-x)=0 \implies 2-x=1 \implies x=1$. This function is decreasing, matches the black curve.

Step2: Analyze $y=1.6\ln(x)$

Domain: $x>0$. It is an increasing logarithmic function, grows faster than standard $\ln(x)$. This matches the green curve.

Step3: Analyze $y=\log(1-x)$

Domain: $1-x>0 \implies x<1$. Find the x-intercept: set $y=0$, $\log(1-x)=0 \implies 1-x=1 \implies x=0$. This function is decreasing, matches the red curve.

Answer:

$y=\ln(2-x)$: c. black
$y=1.6\ln(x)$: a. green
$y=\log(1-x)$: (the remaining red curve, matched as the third option)