Question

f(x) = 2\log_{3}(x) + 6 and g is a vertically scaled version of f. the functions are graphed where f is solid and g is dashed.

what is the equation of g?

choose 1 answer:
a \\( g(x) = \frac{2}{3}\log_{3}(x) + 2 \\)
b \\( g(x) = \log_{3}(x) + 3 \\)
c \\( g(x) = 4\log_{3}(x) + 12 \\)
d \\( g(x) = 6\log_{3}(x) + 18 \\)

Explanation:

Step1: Find point on f(x)

From graph, f(x) has point (1,6). Check: $f(1)=2\log_3(1)+6=0+6=6$. Correct.

Step2: Find corresponding point on g(x)

Corresponding point on g(x) is (1,2).

Step3: Determine vertical scale factor k

Vertical scaling: $g(x)=k\cdot f(x)$? No—wait, vertical scaling of the function form: original $f(x)=2\log_3x +6$, scaled vertically by k gives $g(x)=k(2\log_3x) +6k$? Wait, test x=1: $g(1)=6k=2\Rightarrow k=2/6=1/3$? Wait no—wait option A: at x=1, $g(1)=2/3*0 +2=2$, which matches. Original coefficient of log: 2, option A has 2/3=2(1/3); constant term:6→2=6(1/3). So scale factor is 1/3. Thus $g(x)=\frac{1}{3}(2\log_3x +6)=\frac{2}{3}\log_3x +2$, which is option A.

Answer:

A. $g(x) = \frac{2}{3}\log_{3}(x) + 2$