Earth population growth rate is exponential or logarithmic? How many points on a monotonically increasing curve is needed to determine if it is exponential or logarithmic? For example can we tell that in the most recent history population is increasing exponentially or logarithmicaly?
 A: In a word, neither.
In more detail, there seem to be a number of implicit assumptions in the question (some noted in the comments), including:
1. The earth's population is a well-defined function of time.
In fact, the concept of a "person" has no mathematical definition. For example, birth and death are processes, not instantaneous events.
Of course, with a precise definition a person, the population would be an integer-valued function, while every non-constant continuous function of a real variable fails to be integer-valued.
2. The earth's exact population is a known function of time.
If there were perfect information-gathering, you might conceivably get an exact count of the number of legally alive people at one-day intervals. In reality, census data is nowhere near this comprehensive.
3. The earth's population is a specific function from a finite-parameter family of mathematical functions.
The presumption that the population "is" exponential, or logarithmic, or polynomial (or whatever) conflates a simple, idealized population model for messy reality.
On a related note, functions of one type may very well approximate functions of another type. For instance, every continuous function on a closed, bounded interval can be approximated as closely as you like by polynomials.
4. A mathematical function is uniquely determined by a set of values at finitely many points.
As barrycenter notes in the comments, an arbitrary set of $n$ numerical data is generated by a polynomial of degree at most $n - 1$. Conversely, given four data points, it's likely that no quadratic polynomial or function $p(t) = A + B e^{Ct}$ interpolates them.
5. There is a single, universally-accepted mathematical model for population.
Any population model makes assumptions: The rate of growth is proportional to the population ($p' = kp$), or to the difference between the population and a theoretical "carrying capacity" $M$ ($p' = k(M - p)$), or is jointly proportional to the population and the "amount of room" ($p' = kp(M - p)$), to give three of the simplest among many, many others.
When one "models the world population", one typically starts with census data or other estimates of "actual" population values, then looks for a function of a particular type that in some sense minimizes the total error relative to the data. Ideally, the modeling function should be chosen from a family with "few degrees of freedon"; otherwise one runs the risk of over-fitting, choosing a model whose features are too closely (!) tied to the data.
Population models that are realistic in the sense of the preceding paragraph are probably neither exponential nor logarithmic. The logistic model (the third ODE mentioned two paragraphs up) is fairly common in calculus textbooks, and its solutions are reasonably good qualitative matches for demographic data.
A: @Andrew_D._Hwang's answer is absolutely correct, and these are just 
possibly helpful notes that are too long for a comment. 
The US Census Department has a world population clock: 
http://www.census.gov/popclock/?intcmp=home_pop 
I was hoping they used a formula to compute this (which would tell us 
whether the Census Dept believes growth is exponential, linear, etc), 
but it turns out they don't: 
https://reverseengineering.stackexchange.com/questions/12229 
You can also look at the historical and predicted world population here: 
https://www.census.gov/population/international/data/worldpop/table_population.php 
which fits the line $0.0737374 x-141.441$ (in billions, where x is the 
year number) surprisingly well: 

Of course, this is just putting a straight line through someone else's estimate, not making your own estimate. 
To make your own estimates, you could try various formulas to fit the 
1950-2015 data, or, if you believe the population growth rate has 
changed fundamentally since 1950, a shorter period of time, such as 
2000-2015. 
If you accept the Census' short-term population 
estimates/projections, available in JSON form at: 
https://www.census.gov/popclock/data/population.php/world 
you'll find the line $0.0782816 x-150.481$ fits quite well: 

(note that, to the Census Department, 2015 means the middle of the year 2015). 
although using this line for longer-term historical and projected data doesn't work as well: 
 
So, to a good approximation, you can say the world population is 
growing linearly, both short-term and long-term, but at slightly 
different rates. 
Of course, if you look at any reasonable function over a short 
enough time period, it will look linear, so the analysis above is 
somewhat biased. 
If you want to look longer-term, you might consider using 12,000 years 
worth of estimates: 
https://www.census.gov/population/international/data/worldpop/table_history.php 
Ultimately, it depends on how you define "most recent history" and how 
you plan to use the approximation.
EDIT: Well, since you gave me the checkmark, let me add a couple of things. 
As @Andrew_D._Hwang notes, there are many models of population growth. 
Perhaps the simplest one is the 
https://en.wikipedia.org/wiki/Malthusian_growth_model which assumes 
that, on average, each person alive today will give birth to k 
children. Of course, in our species, only females give birth, but this 
is just an average number of the entire population. 
This model leads to exponential growth, which caused quite a bit of 
concern in the 80s and 90s, since exponential growth is 
unsustainable. Even well-respected science fiction author Isaac Asimov 
was deeply concerned: 
https://asimovfan.wordpress.com/2013/05/09/asimovs-malthusianism/ 
Though it's not in the source above, he believed humanity would fill 
up the galaxy (not just Earth) within 6000 years. 
It turns out this model isn't very realistic. The logistic equation 
@Andrew_D._Hwang mentions 
(https://en.wikipedia.org/wiki/Logistic_function) also known as the 
Verhulst-Pearl equation, postulates that any time two people meet, 
there is a constant, small, but non-zero chance that one will kill the 
other. 
The number of such encounters is proportional to the total number of 
ways in which 2 people can meet each other, $\frac{1}{2} (p-1) p$, 
which is about $\frac{p^2}{2}$. 
In this equation, the population still increases by the birth rate 
above, but also decreases by $b p^2$ due to people killing each other, 
for some constant $b$. 
This Verhulst-Pearl equation ultimately yields a constant population 
with zero growth. 
I don't know of any model that shows long-term linear growth (but that 
might just be me: when I was growing up, the two models above were the 
most popular), so the linear model really is short-term and won't 
last. 
Of course, the Sun will ultimately go nova, after which the Earth's 
living human population will probably be 0, so, in some sense, it's 
all a matter of time scales. 
Other resources: 
http://www.zo.utexas.edu/courses/Thoc/PopGrowth.html 
http://www.sosmath.com/diffeq/first/application/population/population.html 
http://www.nature.com/scitable/knowledge/library/how-populations-grow-the-exponential-and-logistic-13240157 
