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Logarithms for Analog Circuits
16 questions By Tony R. Kuphaldt
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Question 4 of 16
Note the following logarithmic identities, using the “common” (base 10) logarithm:
log10 = 1 log100 = 2 log1000 = 3 log10000 = 4 In the first equation, the numbers 10 and 1 were related together by the log function. In the second equation, the numbers 100 and 2 were related together by the same log function, and so on.
Rewrite the four equations together in such a way that the same numbers are related to each other, but without writing “log”. In other words, represent the same mathematical relationships using some mathematical function other than the common logarithm function.
Reveal answer101 = 10 102 = 100 103 = 1000 104 = 10000 Notes:An illustration like this helps students comprehend what the “log” function actually does.
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Question 5 of 16
Note the following logarithmic identities, using the “common” (base 10) logarithm:
log0.1 = −1 log0.01 = −2 log0.001 = −3 log0.0001 = −4 In the first equation, the numbers 0.1 and 1 were related together by the log function. In the second equation, the numbers 0.01 and 2 were related together by the same log function, and so on.
Rewrite the four equations together in such a way that the same numbers are related to each other, but without writing “log”. In other words, represent the same mathematical relationships using some mathematical function other than the common logarithm function.
Reveal answer10−1 = 0.1 10−2 = 0.01 10−3 = 0.001 10−4 = 0.0001 Notes:An illustration like this helps students comprehend what the “log” function actually does.
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Question 6 of 16
Examine the following progression of mathematical statements:
(102)(103) = 100000 102 + 3 = 100000 105 = 100000 What does this pattern indicate? What principle of algebra is illustrated by these three equations?
Next, examine this progression of mathematical statements:
log105 = log100000 = 5 log102 + 3 = log100000 = 5 log102 + log103 = log100000 = 5 What does this pattern indicate? What principle of algebra is illustrated by these three equations?
Reveal answerFirst pattern:
The product of two base numbers with different exponents is equal to that base number raised to the power of the exponents’ sum.
Second pattern:
The sum of two logarithms is equal to the logarithm of those two numbers’ product.
Notes:In this question, I want students to begin to see how logarithms relate multiplication to addition, and how powers relate addition to multiplication. This is an initial step to students recognizing logarithms as transform functions: a means to transform one type of mathematical problem into a simpler type of mathematical problem.
