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Exponentials and Logarithms

Time constant vs. time.

In the last post, Our First Look at Thermistor details, I talked a little about the exponential increase in resistance as the thermistor was exposed to cold temperatures.  Since then I realized we have not talked in detail what an exponential means.  Also there is more than one base number for exponents and that is also smacking us in the face. so as much as I hate doing it we are going to have to wade into doing a pure math post.  I promise I will make it as easy as I know how.

First, I would like to take a little step aside and talk about something else that is also important.  This site is about technology, but technology is only viewed as a tool to get the job done: to make what we create.  I much prefer to talk about what was called in engineering school “first principles”.  That is the basic science principles and then we will use technology to implement those.  I have found on the web, as well as in real life, many people that love the technology and then find excuses to use it.  The story is that if the tool you know is a hammer then you will see all problems as nails.

For example, in the 1970’s NASA built a large wind turbine.  One of the problems with a wind turbine is if it gets too much wind and basically shakes and spins itself apart.  The story as told to me is much money was spent to analyze the vibrations and if too great, throw a switch to set the pitch of the blades to stop the turbine.  They then looked at how some European engineers solved the same problem thirty years earlier.  Those engineers placed a steel ball tied to a string in a saucer and if the saucer shook too much the ball would fall out, and throw the switch attached to the other end of the string.  I fell into a similar situation on a project where I was thinking of high priced sensors, and the problem was solved with a tennis ball attached to a string.

We will probably use an Adruino as a tool, but unless we find other projects where it will be useful, I will not be dedicating much effort to it.  There are lots and lots of sites dedicated to just the Adruino and I will try to find the better ones and place links to those in the resources.

exponents of base 10 and natural base (e)

Ok, back to the job at hand, exponential and logarithms.  In normal, base 10, the exponent tells us how many times 10 is multiplied by itself.  That simplified answer is correct but does not “make sense” on all the ways exponential is used.  For example, what does 10^3.5 mean?  How do you multiply 10 by itself 1/2 time?  To tell you the truth, I don’t know.  I just know it works and the answer 10^3.5 is 3162.27766.  How do I know that?  I just entered it into a spreadsheet and found the answer.  In that example 3.5 is called the exponent.  In normal text the 3.5 would be written as a superscript (small numbers above and to the right of the 10).  However, this web publishing software does not have superscripts so I wrote it in the method that most computer programming languages and spreadsheets use.  This is shown in the top 1/2 of the 2nd picture in the 2nd column.  (If you click on the picture it will display full size, and use the return button of your browser to return to this page.)

Another not intuitive thing about exponents is negative exponents.  How do you multiply 10 by itself -3 times?   Again, I do not know the how, but I know how it is used. When the exponent is negative, it means 1 divided by the answer if the exponent was positive.  So 10^-3 means 1/10^3 = 1/1000 or 0.001.   The final peculiar case is the exponent of 0.  The answer to that is 1.   This will be true no matter what base we use.  Anything to the power of 0 is always 1.

Related to exponents is scientific numbers.  Often when dealing with very big numbers or very small numbers scientists use a short hand of writing the number and multiplying it by 10 to a power.  This is exactly what we did when we talked about the 3rd band of the resistor color codes in the post “Building and testing the simple amplifier Op-AMP Circuits“.  In spreadsheets and computer programs this is shown by the number followed by E and the exponent.  Again examples are shown in the 2nd picture.  This can come in handy sometimes when doing electrical problems.  For example a 33 K resistor can be inputted as 33E3.

Now that I have filled your head with exponents of base 10, we need to talk about a special base called the natural base or base e.   This has special significance because it is very useful in lots of mathematical functions.  We have already ran into it once when I talked about the thermal time constant of the thermistor.  e is a constant just like π is and just like π it a irrational number, meaning that we can keep finding more and more numbers after the decimal point to get closer to an exact number.  The approximate value of e is 2.71828.  When used in text form the exponential is written e followed by a superscript X where X is the exponent.  In computer programs and spreadsheets it is found by the function exp(X).  (See examples in picture 2).  In picture 1, I show the equation for a time constant for an increasing value.  This shows how the temperature indicated by the thermistor would increase with time if taken from a cold solution to a warm solution and in that case the time axis would be in 20 second increments.

Log(x) & Ln(x)

The inverse of multiply is divide and the inverse of add is subtract, so we should suspect that there is an “undo” of exponents.  There is and it is called the logarithm.  The logarithm of base 10 is called log base 10 or simply log and log(X) is the formula used in spreadsheets and computer programs.   The logarithm of base e is called the natural log and ln(X) is the formula used.  In the third picture I show and example of both the natural exponent and the inverse as well as the same for base 10.

In summary, these are very useful functions to explain many natural occurring processes.  Examples are almost any process that doubles or decreases with a half-life and the discharge or charging of energy storage processes.  I don’t know this, but I would suspect that the effect of drugs entering and exiting our bodies probably follows these same functions.   The functions are hard to calculate by hand but with spreadsheets and calculators they are no problem.

Aside: before calculators, slide rules were based upon logarithms…. but that an ancient history to most now.


Again, as always feel free to e-mail me if you have any questions about this post.  If you find this is tickling your brain cells and enjoy the feeling, please consider subscribing to this blog.



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