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Logarithms for Analog Circuits
16 questions By Tony R. Kuphaldt
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Question 10 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):
12 ÷3 = ??? 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:
- 122 ÷35 =
- 781 ÷92 =
- 19.4 ÷60 =
- 3.5 ÷0.21 =
Reveal answerHere I will show you the steps to using logarithms to solve the first multiplication problem:
12 ÷3 = ??? 12 ÷3 = 10log12 − log3 12 ÷3 = 101.0792 − 0.4771 12 ÷3 = 100.6021 12 ÷3 = 4 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:
12 ÷3 = ??? 12 ÷3 = eln12 − ln3 12 ÷3 = e2.4849 − 1.0986 12 ÷3 = e1.3863 12 ÷3 = 4 -
Question 11 of 16
Examine this progression of mathematical statements:
(1000)2 = 1000000 (1000)2 = 106 log[(1000)2] = log106 (2)(log1000) = log106 (2)(log103) = log106 (2)(3) = 6 What began as an exponential problem ended up as a multiplication 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 exponentiation, down to multiplication, indicates its usefulness as a tool to simplify arithmetic problems. Specifically, the logarithm of a number raised to a power is equal to that power multiplied by the logarithm of the number.
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 12 of 16
Suppose you owned a scientific calculator with two broken buttons: the power (yx) and root (x
√{y}). Demonstrate how you could solve this simple power problem using only logarithms, multiplication, and antilogarithms (powers):
34 = ??? 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:
- 256 =
- 5643 =
- 0.2242 =
- 410.3 =
Reveal answerHere I will show you the steps to using logarithms to solve the first multiplication problem:
34 = ??? 34 = 10(4 log3) 34 = 10(4)(0.4771) 34 = 101.9085 34 = 81 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:
34 = ??? 34 = e(4 ln3) 34 = e(4)(1.0986) 34 = e4.3944 34 = 81
