Logarithmic Double Y Axis Graph Paper: Fill & Download for Free

It is most common to see two-dimensional graphs with both the X and Y axes with linear scales, e.g. 0, 1, 2, 3, …But for a phenomenon where a variable increases exponentially over time, there are some problems plotting it. Here’s an example a variable increasing 10-fold each weekWeek (X axis): 1, 2, 3, 4, 5, 6Variable(Y axis):3, 30, 300, 3,000, 30,000, 3,000,000If that is plotted on regular graph paper, there is no problem with the X-axis. But what about the variable on the Y axis. If you go one division up for the 1st point, then 10 times as far up for the 2nd point, and 100 times as far up for the 3rd point - you’re probably getting off the top of the graph paper.On the other hand, if you mark the top of the graph paper as 3,000,000 then the first 2 or 3 point will all be right at the bottom and not even distinguishable.So instead, we’ll use the Y axis on a Logarithmic scale, so that if one division up represents 3, then the next division up represents 30, then one more division up represents 300, …Now, not only will all the points/data stay on the graph, but there is one more very important additional feature. The points will fall on a straight line, and the slope of the line will give the constant rate of exponential increase. From that you can easily see what is, e.g. “the doubling time” and see that it stays the same over the whole span of data which is plotted.Also if the 7 week value is perhaps 4,000,000, that would show as a large increase on a linear graph, but on the logarithmic graph it would show as a substantial decrease in the earlier exponential rate of increase.That’s why epidemic/pandemic data which shows an exponential rate of increase (at least during the early portion) is often plotted on what is called a semi-logarithmic plot (i.e. semi, because it’s only logarithmic on the Y axis.)

The most common non-linear scale by far is a logarithmic Y. A logarithmic scale means that you have ticks at [math]a^N[/math] , where [math]a[/math] is your base, usually 10 or 2, and N is some selection of whole numbers starting at 0 . With 10 as the base, this means you get ticks at 1,10,100,1000 and so on. It’s a design choice if you want to include half-ticks, either physically halfway between each tick (e.g. at [math]10^{1.5}[/math] = 31.6) or numerically halfway at 50 (about [math]10^{1.7}[/math], so 70% of the physical distance from 10 to 100).Logarithmic Y-axes are quite useful if you want to either show something that acts exponentially, or plot multiple correlated things together when they have wildly different scales. Both are, in a way, the same problem: If you data includes both small and huge numbers, and has details at both ends of the scale, then a linear axis will crush the smaller end down to some noise along the x-axis.A slightly silly example of this is the price of bitcoin over time: In the early days, the price was maybe a few cents, and at the highest it was over $10′000. If you want to show a timeline with important events highlighted, then it’s interesting to see that the price went up tenfold after some news article, even if that increase was from $0.10 to $1. With a linear Y, you’d need a massively tall graph to make that visible while also including the top price.Another benefit is that with a logarithmic Y axis, any multiplication by a constant corresponds to the same distance on paper: Doubling from 1 to 2 and doubling from 1 million to 2 million will both move the line up the same number of pixels or millimeters. As a side effect, this means that anything that grows exponentially will look like a straight line - and you can find the time it takes to double (or grow tenfold) with a ruler.

Lag, log, stationary, and death.In lag phase, the cells are gearing up for growth. There is metabolic activity but no increase in numbers.In Log phase, the cells are doubling exponentially. 1 becomes 2, 2 becomes 4, 4 becomes 8, 8 becomes 16, etc. If graphed on semi-log paper with time on the x-axis (linear scale) and cell number on the Y axis (logarithmic scale), growth presents as a straight line, which is why this phase is called “log phase”. Most experiments in the literature have been performed on cells in this stage.In stationary phase, there is no net increase in cell number. Cells are continuing to divide but other cells are dying off at about the same rate. Arguably, bacteria spend most of their lives in this phase, which is why there has been increasing interest in the field in studying cells during this stage.The death phase is rather variable from one species to another, and even within the same species. Death presumably is due to exhaustion of nutrients and exposure to an excess of waste products. One of the reasons we know this is because bacteria can be cultured to live indefinitely, i.e., be immortal, in a “chemostat”, a device that allows insertion of new nutrition and removal of waste products. In this device, bacteria remain in log phase until removed from the device.