Answer
$y=\log_9{x}$
Work Step by Step
The graph contains the point $(3, \frac{1}{2})$.
This means that the x and y-coordinates of this point satisfy the equation $y=\log_a{x}$.
Substitute the x and y values into the given equation to obtain:
$\frac{1}{2} = \log_a{3}$
RECALL:
$y=\log_a{x} \longrightarrow a^y=x$
Use the rule above to obtain:
$\frac{1}{2} = \log_a{3} \longrightarrow a^{\frac{1}{2}} = 3$
Square both sides to obtain:
$(a^{\frac{1}{2}})^2=3^2
\\(a^{\frac{1}{2}})^2=9$
Use the rule $(a^m)^n = a^{mn}$ to obtain:
$a^{\frac{1}{2} \cdot 2} = 9
\\a^1 = 9
\\a=9$
Thus, the equation of the function whose graph is given is $y=\log_9{x}$.
