Question

given the function $f(x)=\log_{5}x$, which of the following functions is the transformation of $f(x)$ right 6 units, up 3 units, stretched vertically by a factor of 2, and reflected across the x-axis?
(5 points)
$\bigcirc \\ g(x)=-\frac{1}{2}\log_{5}(x + 6)-3$
$\bigcirc \\ g(x)=2\log_{5}(-x + 6)-3$
$\bigcirc \\ g(x)=-2\log_{5}(x - 6)+3$
$\bigcirc \\ g(x)=-\log_{5}\left(\frac{1}{2}x - 6\
ight)-3$

Explanation:

Step1: Shift right 6 units

$g_1(x) = \log_5(x-6)$

Step2: Stretch vertically by 2

$g_2(x) = 2\log_5(x-6)$

Step3: Reflect across x-axis

$g_3(x) = -2\log_5(x-6)$

Step4: Shift up 3 units

$g(x) = -2\log_5(x-6)+3$

Answer:

$g(x) = -2\log_5(x - 6) + 3$