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Introduction to Logarithms

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

Summary: This module provides an introduction to concepts related to logarithms.

Exercise 1

On day 0, you have 1 penny. Every day, you double.

  • a. How many pennies do you have on day 10?
  • b. How many pennies do you have on day n n?
  • c. On what day do you have 32 pennies? Before you answer, express this question as an equation, where x x is the variable you want to solve for.
  • d. Now, what is x x?

Exercise 2

A radioactive substance is decaying. There is currently 100 g g of the substance.

  • a. How much substance will there be after 3 half-lives?
  • b. How much substance will there be after n n half-lives?
  • c. After how many half-lives will there be 1 g g of the substance left? Before you answer, express this question as an equation, where x x is the variable you want to solve for.
  • d. Now, what is x x? (Your answer will be approximate.)

In both of the problems above, part (d) required you to invert the normal exponential function. Instead of going from time to amount, it asked you to go from amount to time. (This is what an inverse function does—it goes the other way—remember?)

So let’s go ahead and talk formally about an inverse exponential function. Remember that an inverse function goes backward. If f ( x ) = 2 x f(x)= 2 x turns a 3 into an 8, then f -1 ( x ) f -1 (x) must turn an 8 into a 3.

So, fill in the following table (on the left) with a bunch of x x and y y values for the mysterious inverse function of 2 x 2 x . Pick x x-values that will make for easy y y-values. See if you can find a few x x-values that make y y be 0 or negative numbers!

On the right, fill in x x and y y values for the inverse function of 10 x 10 x .

Table 1
Inverse of 2 x 2 x
xx y y
8 3
   
   
   
   
   
 
Inverse of 10 x 10 x
x x y y
   
   
   
   
   
   

Now, let’s see if we can get a bit of a handle on this type of function.

In some ways, it’s like a square root. xx size 12{ sqrt {x} } {} is the inverse of x 2 x 2 . When you see xx size 12{ sqrt {x} } {} you are really seeing a mathematical question: “What number, squared, gives me x x?”

Now, we have the inverse of 2 x 2 x (which is quite different from x 2 x 2 of course). But this new function is also a question: see if you can figure out what it is. That is, try to write a question that will reliably get me from the left-hand column to the right-hand column in the first table above.

Do the same for the second table above.

Now, come up with a word problem of your own, similar to the first two in this exercise, but related to compound interest.

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