Find domain and range of logarithmic functions

Question

Find the domain and the range of this function $$f(x)= \log\left(\frac{x + 4}{x-4}\right).$$

Answer 1

$$\frac{x+4}{x-4}>0$$

$$\Rightarrow (x+4>0\land x-4>0)\lor(x+4<0\land x-4<0)$$

$$\Rightarrow x<-4~\lor~x>4$$

$$\Rightarrow x\in(-\infty,-4)\cup(4,\infty)$$

For the range, first find the inverse that is

$$x=\log{\frac{y+4}{y-4}}$$

I suppose it's log with base $10$, then we have

$$10^x=\frac{y+4}{y-4}$$

$$10^xy-10^x(4)=y+4$$

$$y(10^x-1)=4(1+10^x)$$

$$y=\frac{4(1+10^x)}{10^x-1}$$

Then $f(x)$ isn't defined iff $10^x-1=0$, you can finishi the rest

Answer 2

Hints.

What do you know about the argument of the real logarithm?

For the range, set $f(x)=y$ and express $x$ in terms of $y.$ Then find the domain of $x=g(y).$