# Bacterial growth - logarithmic vs. exponential?

November 13, 2013 8:59 AM Subscribe

Why is the log phase of bacterial growth called the log phase, when it actually resembles an exponential curve rather than a logarithmic curve? I'm really dusty on my math, can someone explain this to me?

Exponentials and logarithms are just two sides of the same coin. Using base 10, log(10^2) = 2, log(10^3) = 3, etc. So on a logarithmic scale, exponential growth shows up as a straight line of constant slope. It's very handy.

If it helps to see it visually, look at this graph where the logarithm curve and exponential curve are shown as mirrored about y=x.

posted by wnissen at 9:15 AM on November 13, 2013

If it helps to see it visually, look at this graph where the logarithm curve and exponential curve are shown as mirrored about y=x.

posted by wnissen at 9:15 AM on November 13, 2013

Log phase is also called exponential phase.

posted by florencetnoa at 9:18 AM on November 13, 2013 [2 favorites]

posted by florencetnoa at 9:18 AM on November 13, 2013 [2 favorites]

Best answer: Because it's a straight line if the cell count is plotted on a logarithmic scale. When plotted on a normal scale, it's usually just called the exponential phase.

Also, maybe it just sticks in the mind: Lag,

posted by HFSH at 9:20 AM on November 13, 2013 [1 favorite]

Also, maybe it just sticks in the mind: Lag,

*log*, stationary, death.posted by HFSH at 9:20 AM on November 13, 2013 [1 favorite]

Response by poster: OK. When you're describing something that will grow exponentially, is it just as appropriate to say it will growth logarithmically?

e.g., "Invest your money here and watch it grow exponentially!" --> Would "watch it grow logarithmically!" be just as enticing?

I'm having a discussion with someone about syntax, but since neither of us are familiar with the math, the conversation is going nowhere.

posted by jolyn at 9:30 AM on November 13, 2013

e.g., "Invest your money here and watch it grow exponentially!" --> Would "watch it grow logarithmically!" be just as enticing?

I'm having a discussion with someone about syntax, but since neither of us are familiar with the math, the conversation is going nowhere.

posted by jolyn at 9:30 AM on November 13, 2013

*e.g., "Invest your money here and watch it grow exponentially!" --> Would "watch it grow logarithmically!" be just as enticing?*

Probably not as enticing as most lay-people have no idea what "logarithmically" means (or possibly more enticing since people don't know what the word means).

But the difference is based on your graph scale not on the data so either works in your example as well.

posted by magnetsphere at 9:58 AM on November 13, 2013

Best answer: It's exponential growth, but logarithmic frame of mind is easiest to see it. So, the difference is a matter of frame of reference. Something like the difference between "teaching" and "learning".

posted by notsnot at 9:59 AM on November 13, 2013

posted by notsnot at 9:59 AM on November 13, 2013

Best answer: Log and exponents are kind of the reverse of each other the way that multiplication and division are the reverse; together, they enable you to end up where you started. Which is why log graph paper is called that--you can plot out an exponential function and get a line. This is handy for researchers because the interesting stuff is either on the edges (hard to see on non-log paper for exponential functions that turn into other things) or in the exponential function itself (much easier to calculate from a straight line than a curve.) The population is growing exponentially, which is why it's also called the exponential phase. I wouldn't describe it as growing logarithmically, because that's a different function--"log phase" means "is easier to see this phase on log graph paper", not "phase that follows a log function."

posted by tchemgrrl at 10:05 AM on November 13, 2013 [1 favorite]

posted by tchemgrrl at 10:05 AM on November 13, 2013 [1 favorite]

Best answer: Just to be clear, yes, it's at least confusing, if not actually incorrect. All the explanations you're getting are reasons why it might have come about, not good reasons for keeping it. Most areas of science have terms that got set a long while back, and people haven't bothered to rationalize them.

posted by benito.strauss at 10:07 AM on November 13, 2013 [1 favorite]

posted by benito.strauss at 10:07 AM on November 13, 2013 [1 favorite]

A lot of natural processes are logarithmic in nature. Logarithms make constant proportions a constant distance. For example, every half-life change from 8 to 4 to 2 to 1 to 0.5 and on will be the same distance on a logarithmic plot but change from a between distance of 4, 2, 1, 0.5 on a rectilinear plot. Similarly, growth with a constant doubling time is logarithmic.

Bacterial growth is logarithmic because the doubling time stays constant - until at least other forces take hold.

posted by dances_with_sneetches at 10:13 AM on November 13, 2013

Bacterial growth is logarithmic because the doubling time stays constant - until at least other forces take hold.

posted by dances_with_sneetches at 10:13 AM on November 13, 2013

Response by poster: Very nice. Thanks everyone. Lots to think about here, but I think I'm understanding. The 'log phase' nomenclature makes sense now. I was aware it was also called the exponential phase, but wasn't making the connection. Also, I guess either advertisement would be technically enticing, but the latter would require an understanding of the 'graph scale'. I guess it would be pretty exciting for the person whose retirement portfolio necessitated being plotted on log paper.

posted by jolyn at 11:25 AM on November 13, 2013

posted by jolyn at 11:25 AM on November 13, 2013

*OK. When you're describing something that will grow exponentially, is it just as appropriate to say it will growth logarithmically?*

No. Take your money example, pick a time t in the future (say 10 units of time). Calculate the amount of money you will have by using exponential (exp(10) = 22026) vs. logarithmic (ln(10) = 2.3). Which would you prefer?

posted by doctord at 11:30 AM on November 13, 2013

Response by poster: I would take e^10 obviously.

How do I make the connection to the concept of growth?

So I was still confused in my last post. I was mixing up the concept of logarithmic growth with the concept of graphing something on a logarithmic scale?

Wikipedia says logarithmic growth is "very slow."

posted by jolyn at 12:13 PM on November 13, 2013

How do I make the connection to the concept of growth?

So I was still confused in my last post. I was mixing up the concept of logarithmic growth with the concept of graphing something on a logarithmic scale?

Wikipedia says logarithmic growth is "very slow."

posted by jolyn at 12:13 PM on November 13, 2013

Best answer: Yes, plotting something on a logarithmic scale so that exponential growth appears as a straight line is not the same as logarithmic growth.

Again, using t as time, then you can have different "growth" functions:

exponential ~ exp(t)

linear ~ t

quadratic ~ t^2

cubic ~ t^3

logarithmic ~ ln(t)

etc.

posted by doctord at 12:28 PM on November 13, 2013

Again, using t as time, then you can have different "growth" functions:

exponential ~ exp(t)

linear ~ t

quadratic ~ t^2

cubic ~ t^3

logarithmic ~ ln(t)

etc.

posted by doctord at 12:28 PM on November 13, 2013

Best answer: I believe the misnomer arose because the prevailing equation of cell growth contained a logarithmic term rather than an exponential term. Cell growth rates started receiving a lot of interest around 1910-1920. (See, for example, Slater's "The rate of growth of bacteria,"

posted by Mapes at 2:00 PM on November 13, 2013

*J Chem Soc Trans*(1916)). The predominant observable in this context was a bacteria count or density. The notable finding was that one could take the logarithm of this count, divide it by time, and obtain a growth parameter that was nearly constant. (As Slater expressed it,*k*= 1/*t*log((*N*+*n*)/*n*), where the number of cells increases from*n*to*N*+*n*in time*t*.) It was around this time that it seems the "log growth" nomenclature got entrenched, as the result of using an equation to obtain a biophysical constant from an experimental observation, rather than the other way around.posted by Mapes at 2:00 PM on November 13, 2013

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posted by notsnot at 9:15 AM on November 13, 2013 [2 favorites]