How to draw a graph $f(x)=-\log_{3}(3(x-1))$?

Question

How to draw a graph $f(x)=-\log_{3}(3(x-1))$?

What's with that 3 before x, should I expose it? so I get $f(x)=-\log_{3}(3(x-1))$?

So how do I then draw it? The process (of how I would draw this function):

1.)$f(x)=\log_{3}x$

2.)$f(x)=\log_{3}3x$

3.)$f(x)=\log_{3}(3(x-1))$

4.)$f(x)=-\log_{3}(3(x-1))$

Answer 1

You may want to use the following:

• There is a vertical asymptote when the argument of the $\log$ is zero ($x = 1$).
• The function crosses zero when the argument of the $\log$ is one ($x = 4/3$).
• For large enough $x$, the function behaves like $-\log_3(3x) = -1-\log_3(x)$

Answer 2

Usually we draw this kind of elementary functions by mathematical softwares. On the other hand, your process is correct and feasible. It helps understand the rough figure of $f$.

Answer 3

Let me give you a summary of what you should do. Firstly you should know that the logarithm function is defined only for positive values. Therefore $3(x-1)>0\rightarrow x>1$. When $f(x)=0$ we have, $3x-3=1\rightarrow x=\frac{4}{3}$. Now we shall consider the derivative of this function.

$$f'(x)=-\frac{1}{(x-1)\ln 3}$$

$f'(x)<0$ for all $x>1$.

We also see that, $x\rightarrow \infty\Rightarrow f(x)\rightarrow-\infty$ and when $x\rightarrow 1^+\Rightarrow f'(x)\rightarrow\infty$.

With all the above information we can draw the graph of $f$ and it turns out like >>this<<.