TABLE OF CONTENTS

3.1  Introduction to Logarithms

Goals:

·         Figure out the meaning of logarithms

·         Evaluate logarithms without the use of a calculator

·         Write certain logarithmic equations as exponential equations

·         Write certain exponential equations as logarithmic equations

Prerequisite skills and knowledge:

·         a working knowledge of positive and negative integer exponents

·         a working knowledge of positive and negative rational exponents

Terms to know:

·         integer

·         power

Concept Prep:  Introduction to logarithms

Dynagraph Lab 2 Activities

Example 1.   Find log₁₀(100).  This is sometimes written log (100). 

                        What exponent do we need to raise 10 to                       10^?? = 100

                        to get 100?

                        Since 10² = 100, the log is 2.                                log(100) = 2

Example 2.   Find log(1000)                       

                        What exponent do we need to raise 10 to                       10^?? = 1000

                        to get 1000?

                        Since 10³ = 1000, the log is 3.                              log(1000) = 3

Example 3.   Find log (0.1)   

                        What exponent do we need to raise 10 to                       10^?? =  

                        to get 0.1?

                        Since 10^-1 = 0.1, the log is -1.                               log  = -1.

Example 4.   Find log(200)

                        What exponent do we need to raise 10 to                       10^?? = 200  

                        to get 200?

                        Since 10² = 100 and  10³ = 1000 the log               log(200)  2.3010

                        lies between 2 and 3.  We use the LOG key   

                        on our             calculator to find the approximate

                        value.

                      Check:  10^2.3010   199.9862

Example 5.  Find two consecutive integers between which each of the following lies.

a)     

b)    log(0.005)

Example 6.  Find each of the following.

    a)   

                  Since , we know that  lies between 2 and 3. 

                  We use the LOG key on our calculator and

                   round to four decimal places:                                   

            Please note that this number is an APPROXIMATION for .

b)   log(3,921)

           Since , we know that  lies between 3 and 4. 

            We use the LOG key on our calculator and

           round to four decimal places:                                                      .

            Please note that this number is an APPROXIMATION for .

c)   

            We know from example 6b) above that . 

           We use the LOG key on our calculator and

            round to four decimal places:                                                        .  

      Please note that this number is an APPROXIMATION for .

Logarithms to the base 10 were commonly used years ago to multiply and divide large numbers.   Since logarithms are exponents, we can simply ADD them together rather than multiplying the original numbers.  Of course, we would need a table or calculator to find the logarithms of the original numbers and then to switch back from the logarithms when we are done.   Click here for an example.

      Checkpoint: Introduction to Logarithms

More worked examples

Homework problems

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