About Exponents and Logarithms TABLE OF CONTENTS About Exponents and Logarithms .................................................................................. 1 What is an EXPONENT?................................................................................................ 1 What is a LOGARITHM?............................................................................................... 1 Power Function ................................................................................................................. 1 Squares............................................................................................................................ 1 Calculator Usage......................................................................................................... 2 Cubes............................................................................................................................... 2 Square/Cube Roots ........................................................................................................... 2 Square Roots ................................................................................................................... 2 Cube Roots...................................................................................................................... 3 Exponents .......................................................................................................................... 4 Exponents........................................................................................................................ 4 Properties ........................................................................................................................ 5 Exponential Functions...................................................................................................... 6 Exponential Functions .................................................................................................... 6 Logarithms ........................................................................................................................ 7 Logarithms ...................................................................................................................... 7 Logarithmic Properties.................................................................................................... 8 Changing Bases............................................................................................................... 8 Natural Logarithms .......................................................................................................... 9 Natural Logarithms ......................................................................................................... 9 Glossary ........................................................................................................................... 10 References........................................................................................................................ 13 About Exponents and Logarithms What is an EXPONENT? ● An exponent is an indicator of the number of times a value multiplies itself, denoted by a superscript number placed on the right of the value. ● Exponentiation is used in a wide-range of fields, from mathematics to biology. What is a LOGARITHM? ● A logarithm is the inverse of an exponential function. ● An understanding of logarithms is fundamental to a large number of scientific concepts. ● In mathematics, a logarithm is the power to which a base must be 3 raised to give a specified value (e.g. log 28, is 3, as 2 = 8). Power Function Squares ● The square of x is written as x². It is called a square because it represents the area of a square with a side length of x. As such, when you multiply a number by itself, you get the square of the number. This value will always be non-negative. ● You should know the squares of the integers from 1 to 10. You should also be able to square numbers consisting of an integer followed by zeroes. To do so, simply square the leading integer, then double the number of zeroes on the end. ● It is possible to square fractions, as well, by simply squaring both the numerator and the denominator. 1 ● You can also square numbers with decimal points. Similar to numbers with trailing zeroes, for decimals, there are double the number of terms after the decimal point in comparison with the initial value. The squaring of the term itself is equivalent to squaring the term without regarding the decimals, then accounting for the decimals afterwards. Calculator Usage: ● To square a number with a calculator, look for a button marked x 2. Sometimes this is printed above the key rather than on the key. If this is the case, you will have to press some other button first. It might be labeled INV or 2ND. Most calculators will colour-code the buttons, so you can find the right one easily. IMPORTANT: When you square a number with many digits following the decimal point, using a calculator, you will get an approximation rather than an exact answer. For example, a calculator which displays 10 digits will tell you that 1.00000008 2 = 1.00000016 when it actually equals 1.0000001600000064. Also, when you square numbers, they often become very large or very small, so your calculator will display them using scientific notation. Cubes ● The cube of x is written as x 3. It is called a cube, because it is the volume of a cube with a side length of x. The cube of a number is equivalent to the number times its square, which can be either positive or negative. Square/Cube Roots Square Roots ● The square root is the inverse function of the square function, written as . ● Being the inverse means that if a 2 = b, then p b = a. Geometrically, the inverse of a function is its reflection across the line y = x. The domain of a function is the range of its inverse, and the range of a function is the domain of its inverse. 2 So, since the square of a number is always positive (A positive number times a positive number is positive, and a negative number times a negative number is positive), the range of the square function is all positive real numbers and zero; hence, the domain of the square root function is all positive real numbers and zero. ● Recall, you cannot take the square root of a negative number solely using real numbers. ● Every number has two possible square roots, since: a² = (-a)². Cube Roots ● The cube root function is the inverse of the cube function, written as . 3 ● Cubes can be negative, so you can take the cube root of a negative number, yielding one real root. Exponents Exponents ● The concept of squares and cubes can be generalized to the multiplication of a term by itself any number of times. ● We say "a to the x" or "a to the power of x," when referring to an exponential term. If the exponent is negative, we say "a to the negative x" or "a to the power of negative x." ● The addition of exponents corresponds to multiplication, as shown below, while the subtraction of exponents corresponds to division, in which the exponent of the denominator is subtracted from the exponent of the numerator. Note that all terms used must have a common base. 4 ● As illustrated below, any term to the power of zero is equal to 1. ● The following illustrates properties of rational exponents. ● Where r is rational, and x is irrational, the following holds. Properties 5 Exponential Functions Exponential Functions ● An exponential function is a function of the form y = a x, where a is any positive real number. ● An example of an exponential function is the function y = e x, where e is Euler's number, an irrational number approximately equal to 2.718281828. It is sometimes written as y = exp(x). It is important to distinguish exponential functions from power functions (like square and cube), which are of the form y = x n, where n is a real number. Exponential functions have a constant base and a variable exponent; power functions have a variable base and a constant exponent. ● Exponential functions only take on positive values. ● If a > 1, then the function goes to infinity as x gets larger and to 0 as x gets smaller. ● If 0 < a < 1, then the function goes to 0 as x gets larger and to infinity as x gets smaller. ● All exponential functions pass through the point (0,1). 6 ● A value that grows according to an exponential function is said to exhibit exponential growth. In colloquial speech, this often just means that something is growing fast. Don't confuse this with the precise mathematical definition. If a is just a little bigger than 1, the growth isn't actually that fast. A good way to think of it is: something has exponential growth if it grows faster the longer it has been growing. Logarithms Logarithms ● Logarithmic functions are the inverses of exponential functions. This means that a logarithm function is the reflection of an exponential function across the line y = x. This tells us some important facts: ► Logarithm functions always pass through (1,0). ► Logarithm functions are only defined for positive real numbers. 7 ► For base a > 1 the value of a logarithm gets smaller as x gets closer to 0 and larger as x gets larger. ► For 0 < a < 1 the value gets smaller as x gets larger and larger as x approaches 0. y ● y = log ax if and only if a = x. ● Logarithms are useful for measuring quantities that vary over a large range of values. Logarithmic Properties ● Property 1: log aMN = log aM + log aN ● Property 2: log aM/N = log aM - log aN N ● Property 3: log aM = N x log aM ● Property 4: log a1 = 0 ● Property 5: log aa = 1 ● Property