Exponential & Logarithmic Parent Function 

Consider the parent function $𝑓(𝑥) = \log3x$. 

- vertically compressed by a factor of 1/4   $\Rightarrow$ $𝑓_1(𝑥) = \frac{1}{4}\log3x$

- reflected about the x-axis $\Rightarrow$ $𝑓_2(𝑥) = -\frac{1}{4}\log3x$

- horizontally translated horizontally c units $\Rightarrow$ $𝑓_3(𝑥) = -\frac{1}{4}\log3 (x-c)$

- vertically translated 4 units up $\Rightarrow$ $𝑓_4(𝑥) = -\frac{1}{4}\log3 (x-c)+4$


(a) Since the y-intercept is 3 we have 

\[3=𝑓_4(0) = -\frac{1}{4}\log3 (0-c)+4\]
so 
\[-\frac{1}{4}\log (-3 c)=3-4=-1   \Rightarrow   \log (-3c)=4\]
\[-\frac{1}{4}\log (-3 c)=3-4=-1   \Rightarrow   \log (-3c)=4\]
Hence
\[-3c=e^4.\]
So $$c=\frac{e^4}{-3}.$$

(b) We have
$$𝑓_4(𝑥) = -\frac{1}{4}\log3 (x-c)+4=-\frac{1}{4}\log(3 x+e^4)+4.$$
So we should have $3x+e^4 >0$, and hence
\[x> -\frac{e^4}{3}.\]
So the domain of the function is $(-\frac{e^4}{3}, \infty)$.