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Homework: What Are Logarithms Good For, Anyway?

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides practice problems which develop concepts related to real world applications of logarithms.

Exercise 1

I invest $300 in a bank that pays 5% interest, compounded annually. So after t t years, I have 300 ( 1.05 ) t 300(1.05 ) t dollars in the bank. When I come back, I find that my account is worth $1000. How many years has it been? Your answer will not be a number—it will be a formula with a log in it.

Exercise 2

The pH of a substance is given by the formula pH = - log 10 [ H + ] pH=- log 10 [ H + ], if the concentration of Hydrogen ions.

  • a. If the Hydrogen concentration is 1 10000 1 10000 , what is the pH?
  • b. If the Hydrogen concentration is 1 1000000 1 1000000 , what is the pH?
  • c. What happens to the pH every time the Hydrogen concentration divides by 10?

You may have noticed that all our logarithmic functions use the base 10. Because this is so common, it is given a special name: the common log. When you see something like log ( x ) log(x) with no base written at all, that means the log is 10. (So log ( x ) log(x) is a shorthand way of writing log 10 ( x ) log10(x), just like is a shorthand way of writing . With roots, if you don’t see a little number there, you assume a 2. With logs, you assume a 10.)

Exercise 3

In the space below, write the question that log ( x ) log(x) asks.

Exercise 4

log 100 log100

Exercise 5

log 1000 log1000

Exercise 6

log 10000 log10000

Exercise 7

log ( 1 with n 0s after it ) log( 1 with n 0s after it )

Exercise 8

log 500 (use the log button on your calculator) log500(use the log button on your calculator)

OK, so the log button on your calculator does common logs, that is, logs base 10.

There is one other log button on your calculator. It is called the "natural log," and it is written ln (which sort of stands for "natural log" only backward— personally, I blame the French).

ln means the log to the base e. What is e? It's a long ugly number—kind of like π only different—it goes on forever and you can only approximate it, but it is somewhere around 2.7. Answer the following question about the natural log.

Exercise 9

ln ( e ) = ln(e)=

Exercise 10

ln ( 1 ) = ln(1)=

Exercise 11

ln ( 0 ) = ln(0)=

Exercise 12

ln ( e 5 ) = ln( e 5 )=

Exercise 13

ln ( 3 ) = ln(3)= (*this is the only one that requires the ln button on your calculator)

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