 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 3, Part B:
Exponents and Logarithms

In This Part: Operations with Exponents | Scientific Notation | Logarithms

John Napier from Scotland invented logarithms in the early 17th century. Napier was not a professional mathematician, but he made many important contributions to mathematics. He invented not only logarithms, but the decimal point as well, and he carved multiplication tables on sticks to simplify the multiplication of multi-digit numbers.

What are logarithms? Basically, they are exponents. In order to use logarithms, you must stipulate both the base and the exponent. Here is one example: Since 23 is equal to 8, we can write in symbols log2 8 = 3, which we read as "log to the base two of 8 is 3." This means that the exponent needed on the base two to get to 8 is 3. When working in base ten, it is not necessary to write the base. For example, log 50 is the same as log10 50.

Before the advent of calculators and computers, logarithms were extremely important because they simplified complex multiplication and division by turning them into simple addition or subtraction, and reduced powers to multiplication. Note 1

Most calculators are programmed with values for base ten (abbreviated LOG) and base e (abbreviated LN) logarithms. The letter e represents the transcendental number that is the base of natural logarithms. The value of e is found by taking the limit of (1 + 1/n)n as n approaches infinity. This gives the value of about 2.718.  Problem B8 a. What is log 100?  This question is asking, what exponent is needed in base ten to get 100? In other words, if 10x = 100, what is x?   Close Tip This question is asking, what exponent is needed in base ten to get 100? In other words, if 10x = 100, what is x?

 b. What is log3 81? c. What is the base ten number whose log is 4? d. What is logb b? Problem B9 a. Estimate the value of log5 50. b. Estimate the value of log3 100.  These logarithms will be decimals. Try to figure out between which integers these fractions will fall.   Close Tip These logarithms will be fractions. Try to figure out between which integers these fractions will fall.   Session 3: Index | Notes | Solutions | Video