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Dealing with Large Numbers
Key Idea: Large scale changes in numerical data can be made more manageable by transforming the
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data using logarithms or plotting the data on log-log or log-linear (semi-log) paper.
ffIn biology, numerical data indicating scale can often decrease or increase exponentially. Examples include the exponential growth of populations, exponential decay of radioisotopes, and the pH scale.
ffExponential changes in numbers are defined by a function. A function is simply a rule that allows us to calculate an output for any given input. Exponential functions are common in biology and may involve very large numbers.
ffLog transformations of exponential numbers can make them easier to handle. Exponential function ffExponential growth occurs at an increasingly rapid
rate in proportion to the growing total number or size.
Log transformations
ff A log transformation makes very large numbers easier to work with. The log of a number is the exponent to which a fixed value (the base) is raised to get that number. So log10 (1000) = 3 because 103 = 1000.
ffIn an exponential function, the base number is fixed (constant) and the exponent is variable.
ffThe equation for an exponential function is y = cx. ffExponential growth and decay (reduction) are possible.
ff Both log10 (common logs) and loge (natural logs or ln) are commonly used.
ff Log transformations are useful for data where there is an
exponential increase or decrease in numbers. In this case, the transformation will produce a straight line plot.
ffExponential changes in numbers are easy to identify
because the curve has a J-shape appearance due to its increasing steepness over time.
ffAn example of exponential growth is the growth of a microbial population in an unlimiting, optimal growth environment.
ff To find the log10 of a number, e.g. 32, using a calculator, key in log 32 = . The answer should be 1.51.
ff Alternatively, the untransformed data can be plotted
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Population number
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Smaller numbers can't be read off the graph
800 400 600 Time (minutes) Example: Cell growth in a yeast culture where growth is not limited by lack of nutrients or build up of toxins. 0
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1. Why is it useful to plot exponential growth using semi-log paper?
Large numbers are easily accommodated
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Smaller numbers are easily read off the graph
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400 600 800 Time (minutes) Example: The same yeast cell growth plotted on a loglinear scale. The y axis present 6 exponential cycles
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Further increase is not easily accommodated
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directly on a log-linear scale (as below). This is not difficult. You just need to remember that the log axis runs in exponential cycles. The paper makes the log for you.
2. What would you do to show yeast exponential growth as a straight line plot on normal graph paper?
3. Log transformations are often used when a value of interest ranges over several orders of magnitude. Can you think of another example of data from the natural world where the data collected might show this behaviour?
Š 2017 BIOZONE International ISBN: 978-1-927309-60-5 Photocopying Prohibited
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