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Logarithms for Analog Circuits
16 questions By Tony R. Kuphaldt
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Question 7 of 16
Examine this progression of mathematical statements:
(100)(1000) = 100000 (100)(1000) = 105 log[(100)(1000)] = log105 log100 + log1000 = log105 log102 + log103 = log105 2 + 3 = 5 What began as a multiplication problem ended up as an addition problem, through the application of logarithms. What does this tell you about the utility of logarithms as an arithmetic tool?
Reveal answerThat logarithms can reduce the complexity of an equation from multiplication, down to addition, indicates its usefulness as a tool to simplify arithmetic problems. Specifically, the logarithm of a product is equal to the sum of the logarithms of the two numbers being multiplied.
Notes:In mathematics, any procedure that reduces a complex type of problem into a simpler type of problem is called a transform function, and logarithms are one of the simplest types of transform functions in existence.
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Question 8 of 16
Suppose you owned a scientific calculator with two broken buttons: the multiply (×) and divide (÷). Demonstrate how you could solve this simple multiplication problem using only logarithms, addition, and antilogarithms (powers):
7 ×5 = ??? The answer to this problem was easy enough for you to figure out without a calculator at all, so here are some more practice problems for you to try:
- 23 ×35 =
- 781 ×92 =
- 19.4 ×60 =
- 0.019 ×2.6 =
Reveal answerHere I will show you the steps to using logarithms to solve the first multiplication problem:
7 ×5 = ??? 7 ×5 = 10log7 + log5 7 ×5 = 100.8451 + 0.6990 7 ×5 = 101.5441 7 ×5 = 35 Since the others are easy enough for you to check (with your non-broken calculator!), I’ll leave their solutions in your capable hands.
Notes:Incidentally, there is nothing special about the common logarithm to warrant its exclusive use in this problem. We could have just as easily applied the natural logarithm function with the same (final) result:
7 ×5 = ??? 7 ×5 = eln7 ln5 7 ×5 = e1.9459 1.6094 7 ×5 = e3.5553 7 ×5 = 35 -
Question 9 of 16
Examine this progression of mathematical statements:
1000 100= 10 1000 100= 101 log ( 1000 100) = log101 log1000 − log100 = log101 log103 − log102 = log101 3 − 2 = 1 What began as a division problem ended up as a subtraction problem, through the application of logarithms. What does this tell you about the utility of logarithms as an arithmetic tool?
Reveal answerThat logarithms can reduce the complexity of an equation from division, down to subtraction, indicates its usefulness as a tool to simplify arithmetic problems. Specifically, the logarithm of a quotient is equal to the difference between the logarithms of the two numbers being divided.
Notes:In mathematics, any procedure that reduces a complex type of problem into a simpler type of problem is called a transform function, and logarithms are one of the simplest types of transform functions in existence.
