Logarithms and Exponents: Partners in Mathematical Harmony

Using Integer Exponents Positive integer exponents are a notation to express a number raised to any given power. They are defined as a superscripted number. The number above the base is calledthe exponent and is used to represent a power of the base. For example, 2^4 = 2 * 2 * 2 *2 = 16 The exponent tells us how many times the base can be multiplied by itself. For example, 2^4 means 2 * 2 * 2 * 2 = 16. Negative integer exponents are defined as a superscripted negative number. The number above the base is called the exponent and is used to represent a power of thebase. For example, (-2)^4 = (-2) * (-2) * (-2) * (-2) = 256. We're going to begin with the basic integer exponents. We'll start with positive integer exponents. Positive integer exponents are symbols used for counting, such as 1, 2, 3 and so on. When we have a number like 9, which is equal to 3 times 3. Three is called a factor ofnine because it occurs twice in the number. In the number 27, the factor 3 occurs 3 times. And of course, in the number 81, the factor 3 occurs 4 times. Positive integer exponents—numbers placed in the upper rightcorner of a factor—can be used to count how many times the same factor repeats in anumber. So, 9 equals 3 times 3 equals 3 to the second power, where 2 is the exponent. 27 equals 3 times 3 equals 3 to the third power, where 3 is the exponent. And 81 equals 3times 3 times 3 equals 3 to the fourth power, where 4 is the exponent and so on. Thus, weread 3 to the 4th as 3 raised to the 4th power or simply 3 to the 4th for short. So now let's look at another example. The number 248,832 is just equal to 12 raised to the 5th power, or 12 to the 5th. This is convenient because it saves us fromhaving to write out every digit of 12 repeatedly. For historical reasons, numbers raise to thesecond power have special names as well as a regular name. Thus, 4 squared is alsocalled 4 to the second power, and 4 cubed is also called 4 to the third power. Next, we will discuss 0 as an exponent. The problems involving zero-exponents are easy; this is because, by definition, any number except 0 raised to the power of 0equals 1. For example, 3 to the 0 equals 1, 2 to the 0 equals 1, 2π to the 0 equals 1, 1/xcubed to the 0 equals 1 as long as x does not equal 0, and so on. For mathematicians,zeros raised to or brought down to negative powers are too weird and are undefined. Next, we'll examine negative integer exponents. 2 to the power of -1 is read as "2 to the minus 1" and is equal to 1 divided by 2 raised to the 1th power or 1/2. 2 to the -2 isequal to 1 over 2 squared or one quarter. 2 to the -3 is equal to 1 over 2 cubed or oneeighth. Raising a number to a negative exponent is equivalent to dividing it by the same integer if that integer were positive. Now let's consider dividing by a negative exponent. Using the same logic we used in the previous paragraph,1/2 to the -2 simply = 2 to the 1 or 2. 1/2 to the -3 simply = 2 to

the 3rd or 2 cubed or 8. We express the general rule for negative exponents using lettersto stand for just about any number. The general rule is x to the n equals 1 over x to the n, and 1 over x to the n equals x to the n. The most common use of exponents is for scientific notation, which is a way towrite numbers that otherwise would have a very large number of zeros in them. We move one digit to the left of the decimal place in the significant number 5,972. This gives us 5.972. We then multiply by 10 raised to the appropriate exponent, which forthis case is 24. So 10 to the 24 means that you are going to move the decimal place 24places to the right. Thus, if a number is less than one, we move the decimal place to the left. If we have an exponent of negative three, we move the decimal place 31 places to the left. Theelectron has a mass of 9.109 x 10–31 kg. We move the decimal 31 places to the left toobtain this number. The key thing to remember about scientific notation is that we only need to keep significant non-zero digits and we always have one digit, to the left of the decimal place.One digit to the left. That's what you need to know for scientific notation.