The general form of a logarithmic function is f(x) = k + alogb(x - h), where a, b, h and k are reals number so that b is a positive number different to 1 and x - h > 0.
Let's suppose we start with the graph of log₁₀(x), if we have k = 3 we are translating this graph 3 units up, and then one transformation of the logarithm function is:
1) f(x) = log₁₀(x) + 4
Similarly, by taking a as 2, we are scalling vertically the graph with the following equation:
2) f(x) = 2log₁₀(x)
By taking h as 1 we are translating the graph 1 unit to the right, like this:
3) f(x) = log₁₀(x - 1)
We can do all these transformation simultaneously, to get:
4) f(x) = 2log₁₀(x - 1) + 4
Similarly, we can generate 21 more transformations, like this:
5) f(x) = log₁₀(x) - 3 (vertical translation 3 units down)
6) f(x) = 10log₁₀(x) + 7 (vertical translation 7 units up and vertical scaling by a factor of 10)
7) f(x) = 2log₁₀(x + 9) (vertical sacling by a factor of 2 and 9 units translation to the left)
8) f(x) = 5log₁₀(x - 5) - 5 (vertical sacling by 5 and 5 units translation to the right and 5 units down)
9) f(x) = log₁₀(x) - 1 (vertical translation 1 unit down)
10) f(x) = 30log₁₀(x) (vertical scaling by a factor of 30)
11) f(x) = 10log₁₀(x + 2)
12) f(x) = 3log₁₀(x) - 1
13) f(x) = (1/2)log₁₀(x) + 7
14) f(x) = log₁₀(x - 18)
15) f(x) = 7log₁₀(x + 3)
16) f(x) = log₁₀(x) - 21
17) f(x) = 2log₁₀(x + 10) - 5
18) f(x) = 4log₁₀(x)
19) f(x) = log₁₀(x - 8)
20) f(x) = log₁₀(x) + 33
21) f(x) = -log₁₀(x)
22) f(x) = -4log₁₀(x) - 3
23) f(x) = 9log₁₀(x) + 8
24) f(x) = -7log₁₀(x + 3)
25) f(x) = 3log₁₀(x - 4) + 11