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Logarithms for Analog Circuits
16 questions By Tony R. Kuphaldt
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Question 1 of 16
The concept of a mathematical power is familiar to most students of algebra. For instance, ten to the third power means this:
103 = 10 ×10 ×10 = 1000 . . . and eight to the seventh power means this:
87 = 8 ×8 ×8 ×8 ×8 ×8 ×8 = 2,097,152 Just as subtraction is the inverse function of addition, and division is the inverse function of multiplication (because with inverse functions, one “undoes” the other), there is also an inverse function for a power and we call it the logarithm.
Re-write the expression 103 = 1000 so that it uses the same quantities (10, 3, and 1000) in the context of a logarithm instead of a power, just as the subtraction is shown here to be the inverse of addition, and division is shown to be the inverse of multiplication in the following examples:
3 + 8 = 11 ( and − are inverse functions) 11 − 3 = 8 2 × 7 = 14 (× and ÷are inverse functions) 14 ÷ 2 = 7 103 = 1000 (powers and logs are inverse functions) log10 ??? = ??? Reveal answer103 = 1000 (powers and logs are inverse functions) log10 1000 = 3 Notes:In my experience, most American students are woefully underprepared for the subject of logarithms when they study with me. Admittedly, logarithms do not see as much use in everyday life as powers do (and that is very little for most people as it is!). Logarithms used to be common fare for secondary school and college students, as they were essential for the operation of a slide rule, an elegant mechanical analog computing device popular decades ago.
The purpose of this question is to twofold: to get students to realize what a logarithm is, and also to remind them of the concept of inverse functions, which become very important in analog computational circuits.
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Question 2 of 16
Given the following mathematical expression, write another one defining a logarithm using the same variables:
If: xy = z Then: log? ? = ? Reveal answerIf: xy = z Then: logx z = y Notes:Nothing special here. Indeed, the answer to this question may be derived from any algebra textbook.
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Question 3 of 16
Electronic calculators with logarithm capability have at least two different types of logarithms: common logarithm and natural logarithm, symbolized as “log” and “ln”, respectively. Explain what the difference is between these two types of logarithms.
Reveal answerThe common logarithm function assumes a “base” value of ten, whereas the natural logarithm assumes a base value of e (Euler’s constant).
Follow-up question: what is the approximate value of e? How can you get your calculator to give you the answer (rather than looking it up in a math book?
Notes:Some calculators, of course, allow you to extract the logarithm of any number to any base. Here, I simply want students to become familiar with the two logarithm functions available on the most basic scientific calculators.
Note that some calculators will show just enough digits of e to give the false impression that they repeat (ten digits: e = 2.718281828). If anyone suggests that e is a (rational) repeating decimal number, correct this misunderstanding by telling them it is irrational just like π.
