Graphing logarithmic functions

Question

Currently, in my class we just started learning about graphing logarithmic functions and I was absent today so all I have is a worksheet and a blank note sheet that doesn't help with anything really.

I've tried watching videos and they helped me with basic graphing of them, but I get very confused once we get to translating them like f(x) = log(x-1)+2. Also, on most of the problems I've seen there's usually a number between the log and the (x-1), like this log2(x-1)+2 instead of what I put above so I don't know what to do.

So how do I graph the translated log functions, like f(x) = log(x-1)+2, any help?

Answer 1

If you take the expression of a function and replace $x$ by $x-1,$ it moves the entire graph of the function $1$ unit to the right. If you replace $x$ with $x-2,$ it's $2$ units to the right. For $x+1$ it's one unit to the left. And so forth.

If you take the final output of a function and add $2$, you move the graph $2$ units upward.

These translation ideas are not special to logarithmic functions. Essentially any function you can graph will move like this when you modify its input or output in these ways.

If you see something like $\log_2,$ the $2$ is the base of the logarithm. All logarithms have a base; if you are writing $\log$ without an indication of the base, it means you are assuming the base is the "usual" one, whatever is "usual" for you. For most mathematics, the "usual" base is $e$ (that is, we use the natural logarithm). But in some contexts the "usual" base is $10,$ which I suppose is why $\log_{10}$ has been known as the common logarithm.

Answer 2

You can ask Alpha to do the plot for you, getting this
The $x-1$ just shifts the graph one unit to the right compared to $\log x$. The $+2$ shifts the graph upward by two units. When you see $\log_2 x$ that is the base $2$ log instead of base $10$ or $e$

Answer 3

I think some experience is good:

Desmos.com

Play around with each thing and record how it affects the logarithm.