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Homework: Properties of Logarithms

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

Summary: This module provides practice problems which develop concepts related to properties of logarithms.

Memorize these three rules of logarithms.

  • log x ( ab ) = log x a + log x b log x (ab)= log x a+ log x b
  • log x ( a b ) = log x a - log x b log x ( a b )= log x a- log x b
  • log x ( a b ) = b log x a log x ( a b )=b log x a

Exercise 1

In class, we demonstrated the first and third rules above. For instance, for the first rule:

log 2 8 = log 2 ( 2 × 2 × 2 ) = 3 log 2 8= log 2 (2×2×2)=3

log 2 16 = log 2 ( 2 × 2 × 2 × 2 ) = 4 log 2 16= log 2 (2×2×2×2)=4

log 2 ( 8 × 16 ) = log 2 [ ( 2 × 2 × 2 ) ( 2 × 2 × 2 × 2 ) ] = 7 log 2 (8×16)= log 2 [(2×2×2)(2×2×2×2)]=7

This demonstrates that when you multiply two numbers, their logs add.

Now, you come up with a similar demonstration of the second rule of logs, that shows why when you divide two numbers, their logs subtract.

Now we’re going to practice applying those three rules. Take my word for these two facts. (You don’t have to memorize them, but you will be using them for this homework.)

  • log 5 8 = 1.29 log 5 8=1.29
  • log 5 60 = 2.54 log 5 60=2.54

Now, use those facts to answer the following questions.

Exercise 2

log 5 480 = log 5 480=

Hint:

480 = 8 × 60 480=8×60. So this is log 5 ( 8 × 60 ) log 5 (8×60). Which rule above helps you rewrite this?

Exercise 3

How can you use your calculator to test your answer to #2? (I’m assuming here that you can’t find log 5 480 log 5 480 on your calculator, but you can do exponents.) Run the test—did it work?

Exercise 4

log 5 215 log 5 215 size 12{ left ( { {2} over {"15"} } right )} {} ==

Exercise 5

log 5 152= log 5 152 size 12{ left ( { {"15"} over {2} } right )} {}=

Exercise 6

log 5 64 = log 5 64=

Exercise 7

log 5 ( 5 ) 23 = log 5 (5 ) 23 =

Exercise 8

5 ( log 5 23 ) = 5 ( log 5 23 ) =

Simplify, using the log ( x y ) log(xy) property:

Exercise 9

log a ( x • x • x • x ) log a (x•x•x•x)

Exercise 10

log a ( x • 1 ) log a (x•1)

Simplify, using the log log xylogxy size 12{ left ( { {x} over {y} } right )} {} property:

Exercise 11

log a 1x log a 1x size 12{ left ( { {1} over {x} } right )} {}

Exercise 12

log a x1 log a x1 size 12{ left ( { {x} over {1} } right )} {}

Exercise 13

log a xx log a xx size 12{ left ( { {x} over {x} } right )} {}

Simplify, using the log ( x ) b log(x ) b property:

Exercise 14

log a ( x ) 4 log a (x ) 4

Exercise 15

log a ( x ) 0 log a (x ) 0

Exercise 16

log a ( x ) -1 log a (x ) -1

Exercise 17

  • a. Draw a graph of y = log 1 2 x y= log 1 2 x. Plot at least 5 points to draw the graph.
  • b. What are the domain and range of the graph? What does that tell you about this function?

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