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The logarithm as an inverse function In this section we concentrate on understanding the logarithm function. If the logarithm is understood as the inverse of the exponential function, then the properties of logarithms will naturally follow from our Elementary Functions understanding of exponents. Part 3, Exponential Functions & Logarithms The meaning of the logarithm. Lecture 3.3a, Logarithms: Basic Properties The logarithmic function g(x) = logb(x) is the inverse of the exponential function f(x) = bx: Dr. Ken W. Smith y The meaning of y = logb(x) is b = x: Sam Houston State University The expression y 2013 b = x is the \exponential form" for the logarithm y = logb(x). The positive constant b is called the base (of the logarithm.) Smith (SHSU) Elementary Functions 2013 1 / 29 Smith (SHSU) Elementary Functions 2013 2 / 29 Logarithms Logarithms Some worked exercises. Write each of the following logarithms in exponential form and then use that exponential form to solve for x. 1 4 log2( 8 ) = x x 1 −3 1 1 log2(8) = x Solution. The exponential form is 2 = 8 : Since 2 = 8 the answer Solution. The exponential form is 2x = 8: Since 23 = 8 the answer is is x = −3 . p x = 3 . 3 5 log ( 2) = x 47 2 p 2 log2(2 ) = x Solution. The exponential form is 2x = 3 2 = 21=3: So x = 1=3 . Solution. The exponential form is 2x = 247: So x = 47 . 1 3 log2( 2 ) = x 1 Notice how we use the exponential form in each problem! Solution. The exponential form is 2x = 1 : Since 2−1 = the answer 2 2 is x = −1 . Smith (SHSU) Elementary Functions 2013 3 / 29 Smith (SHSU) Elementary Functions 2013 4 / 29 The graph of a logarithm function The graph of a logarithm function The graph of y = 2x was drawn in an earlier lecture (see below.) The graph of y = log2 x : The graph of the inverse function y = log2 x is obtained by reflecting the graph of y = 2x across the line y = x: Smith (SHSU) Elementary Functions 2013 5 / 29 Smith (SHSU) Elementary Functions 2013 6 / 29 The graph of a logarithm function The graph of a logarithm function If we draw them together, we have the picture below. The graph of the exponential function y = 2x: The graph of the logarithmic function y = log x: Smith (SHSU) Elementary Functions 2013 7 / 29 Smith (SHSU) Elementary Functions 2 2013 8 / 29 Logarithms Logarithms We agreed earlier that the exponential function f(x) = bx has domain (−∞; 1) and range (0; 1). Since g(x) = logb x is the inverse function of f(x) the domain of the log function will be the range of the exponential function, and vice versa. In summary, here are our abbreviations: So the domain of logb x is (0; 1) and the range is (−∞; 1): 1 ln x means the logarithm base e, The most useful base for logarithms is e. We will abbreviate loge(x) by 2 log x means the logarithm base 10 and ln(x) and speak of the \natural logarithm". 3 lb x means the logarithm base 2. Sometimes, for historical reasons, we may use base 10. It is customary to speak then of the \common logarithm" and abbreviate log10(x) by log(x); dropping the subscript. However (warning!), in higher mathematics and engineering applications, log(x) usually means base e and is equivalent to ln(x): In these notes we will use log(x) to mean log10(x): One more abbreviation { in computer science, because computers store Smith (SHSU) Elementary Functions 2013 9 / 29 Smith (SHSU) Elementary Functions 2013 10 / 29 data in binary (in bits of zeroes and ones), one uses base 2. Some Logarithmsabbreviate log2(x) as lb(x) and speak of the \binary" logarithm. Properties of exponential functions in terms of logarithms The logarithm function plucks the exponent from an expression. For this A few more worked exercises. reason, the properties of exponents translate into properties of logarithms. Write each of the following logarithms in exponential form and then use For example, we know that when we multiply two terms with a common that exponential form to solve for x. base, we add the exponents: 1 log(1000) = x x y x+y Solution. The exponential form is 10x = 1000: Since 103 = 1000 the (b )(b ) = b (1) answer is x = 3 . Suppose we call the first term M := bx and the second term N := by. 1 Then one may ask the question, \What is the exponent on b in the 2 ln( ) = x e3 product MN? Solution. The exponential form is ex = e−3 so the answer is −3 . The answer is \We add the exponents appearing in M and N." In other 1 words (if we learn to translate \logb" as \the exponent on b that..."), we 3 lb(p ) = x 2 can restate this exponent property as \when we multiply numbers we add 1 p their exponents". This is the product property for logarithms: Solution. The exponential form is 2x = p : Since 21=2 = 2 then 2 1 2−1=2 = p and so the answer is x = −1=2 . logb(MN) = logb M + logb N (2) 2 Smith (SHSU) Elementary Functions 2013 11 / 29 Smith (SHSU) Elementary Functions 2013 12 / 29 Logarithms Logarithms x What happens when we divide two terms with a common base? A third important property of exponents: when we raise a term like b to a power, we multiply exponents. x b x−y = b (3) x c xc by (b ) = b (5) When we do division, we subtract exponents. So, in the language of In our \logarithm language" (thinking of M as bx) we have the exponent logarithms, we have the quotient property, \the exponent in a quotient is property the difference of the two exponents": c logb(M ) = c logb M (6) M log ( ) = log M − log N (4) b N b b Each of these three properties is merely a restatement, in the language of logarithms, of a property of exponents. Smith (SHSU) Elementary Functions 2013 13 / 29 Smith (SHSU) Elementary Functions 2013 14 / 29 Logarithms Exponential Functions We review the three basic logarithm rules we have developed so far. Product Property of Logarithms: logb(MN) = logb M + logb N Quotient Property of Logarithms: In the next presentation, we develop several more properties of logarithms. M (END) log ( ) = log M − log N b N b b Exponent Property of Logarithms: c logb(M ) = c logb M Each of these properties is a restatement, in the language of logarithms, of a property of exponents. Smith (SHSU) Elementary Functions 2013 15 / 29 Smith (SHSU) Elementary Functions 2013 16 / 29 Logarithms We review the three basic logarithm rules we have developed so far. Product Property of Logarithms: Elementary Functions logb(MN) = logb M + logb N Part 3, Exponential Functions & Logarithms Lecture 3.3b, Logarithms: Basic Properties, Continued Quotient Property of Logarithms: M log ( ) = log M − log N Dr. Ken W. Smith b N b b Sam Houston State University Exponent Property of Logarithms: 2013 c logb(M ) = c logb M Each of these three properties is merely a restatement of a property of exponents. Smith (SHSU) Elementary Functions 2013 17 / 29 Smith (SHSU) Elementary Functions 2013 18 / 29 Changing the base Changing the base Suppose we want to change the base of our logarithm. This often occurs We began with when we want to use a \good" base like e on a problem which began with y = log x: a different base. b Suppose we want to work with base c but our problem began with base b: We rewrote this as log x y = c : logc b y = logb x: Rewrite this in exponential form: So y, which was originally equal to logb x is now y b = x: logc x logb x = Now take the log of both sides of the equation. If we want to work in base logc b c then let us apply log () to both sides of our equation. c Let's call this the \change of base" equation or \change of base" y logc(b ) = logc(x): property. Now we use the exponent property pulling the exponent y outside the logarithm: One way to remember this is to note that on the left side of the equal sign y log (b) = log (x): (logb x), b is lower than x. c c Then on the right side of the equal sign ( log x ), b is still lower than x! Solve for y: log b log x Smith (SHSU) Elementaryy = Functionsc : 2013 19 / 29 Smith (SHSU) Elementary Functions 2013 20 / 29 logc b Logarithms More on the logarithm as an inverse function We began this lecture by defining g(x) = logb(x) as the inverse function of f(x) = bx. Since these functions are inverses, we know then that (f ◦ g)(x) = (g ◦ f)(x) = x: (7) Example. Suppose we want to compute log (17) but our calculator only 2 Let us examine this in more detail. allows us to use the natural logarithm ln. Then, by the change of base equation we can write x x Note that (g ◦ f)(x) = g(f(x)) = g(b ) = logb(b ): Since the log function ln 17 and the exponential function are inverse functions, this must be equal to log (17) = ≈ 4:087463: just x and we have 2 ln 2 x logb(b ) = x: This equation is really fairly easy to understand. If we translate \logb(x)" x as \the exponent on b that give x" then we should translate logb(b ) as \the exponent on b which gives bx." Obviously this should be x since x is the exponent one places on b to get bx: (If that doesn't make sense, read through it one more time slowly....) Smith (SHSU) Elementary Functions 2013 21 / 29 Smith (SHSU) Elementary Functions 2013 22 / 29 More on the logarithm as an inverse function Six properties of logarithms In summary, we have developed the following six properties of logarithms.
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