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Objectives – Change from logarithmic to exponential form. – Change from exponential to logarithmic form. – Evaluate logarithms. – Use basic logarithmic properties. – Graph G h logarithmic l ith i ffunctions. ti – Find the domain of a logarithmic function. – Use common logarithms. – Use natural logarithms. Logarithmic Functions Section 3.2 Inverse of y = 3x Flashback Consider the graph of the exponential function y = f(x) = 3x. Begin with Interchange variables • Is f(x) one-to-one? • Does f(x) have an i inverse th thatt is i a function? • Find the inverse. y = 3x x = 3y Now, solve for y. y= the th power to t which hi h 3 mustt be b raised i d in i order d to obtain x. Symbolically, y = log 3 x “The logarithm, base 3, of x.” Domain Restrictions for Logarithmic Functions Logarithm For all positive numbers b, where b ≠ 1, b y =x is equivalent to ≠ y=logb x Logbx is an exponent to which the base b must be raised to obtain x. • Since a positive number raised to an exponent (positive or negative) always results in a positive value, you can ONLY take the logarithm of a POSITIVE NUMBER. • Remember, the question is: What POWER can I raise the base to, to get this value? • DOMAIN RESTRICTION: y = logb x, x > 0 1 Write on top of your test!! log a x = y ↔ x = a y A logarithm is an exponent! Common Logarithms Logarithms, base 10, are called common logarithms. For all p positive numbers x, lo g x = lo g 1 0 x Log button on your calculator is the common log. Example • Find each of the following common logarithms on a calculator. Round to four decimal places. a) log 723,456 b) log 0.0000245 c)) log l ((−4) 4) Natural logarithms • Logarithms, base e, are called natural logarithms. • ln(x) represents the natural log of x, which has a base=e x ⎛ 1⎞ • What is e? If you plug large values into ⎜1 + ⎟ yyou g get closer and closer to e. ⎝ x⎠ • logarithmic functions that involve base e are found throughout nature • Calculators have a button “ln” which represents the natural log. log e x = ln( x) Example • Find each of the following natural logarithms on a calculator. Round to four decimal places. a) ln 723,456 b) ln 0.0000245 c) ln (−4) Logarithmic Conversions • Convert each of the following to a logarithmic equation. a) 25 = 5x b) ew = 30 2 Example • Convert each of the following to an exponential equation. Rewrite the following exponential expression as a logarithmic one. 3( x + 2 ) = 7 1) log 7 ( x + 2) = 3 a) log7 343 = 3 2) log 3 ( x + 2) = 7 b) logb R = 12 3) log 3 (7) = x + 2 4) log 3 ( x − 2) = 7 Finding Logarithms • Find each of the following logarithms without using a calculator. a) log2 16 b) log10 1000 c) log16 4 d) log10 0.001 Summary of Properties of Logarithms F o r a > 0 , a ≠ 1 ,a n d a n y re a l n u m b e r k , 1 . lo g a a = 1 , ln e = 1 2 . lo g a 1 = 0 , ln 1 = 0 A d d itio n a l L o g a rith m ic P ro p e rtie s 3 . lo g a a k = k 1 1 = = 10−3. 1000 103 Example Evaluate each expression without using a calculator. a.) log6 6 b.)log11 11 c.)ln1 c )ln1 d.)log15 1 e.)log4 46 4 . a lo g a k = k , k > 0 Logarithmic Functions • Logarithmic functions are inverses of exponential functions. Graph y = log 3 x by converting to its equivalent exponential form: x = 3y 1. Choose values for y. y 2. Compute values for x. 3. Plot the points and connect them with a smooth curve. * Note that the curve does not touch or cross the y-axis. 3 Logarithmic Functions continued Graph: x = 3y y = log 3 x x = 3y y (x, y) 1 0 (1, 0) 3 1 (3 1) (3, 9 2 (9, 2) 1/3 −1 (1/3, −1) 1/9 −2 (1/9, −2) 1/27 −3 (1/27, −3) Comparing Exponential and Logarithmic Functions Logarithmic Functions Remember: Logarithmic functions are inverses of exponential functions. Th iinverse off f(x) The f( ) = a x is given by f -1(x)=loga x Asymptotes • Recall that the horizontal asymptote of the exponential function y = ax is the x-axis. • The horizontal axis of a logarithmic function y=loga x is the y-axis. 4 Graphs of Logarithmic Functions • Graph: y = f(x) = log6 x. – Select y. – Compute x. x, or 6 y 1 6 36 216 1/6 1/36 Transformations of logarithmic functions are treated as other transformations. • Follow order of operations. • Note: When graphing a logarithmic function,, the graph g p only y exists for x>0. • WHY? y 0 1 2 3 −1 −2 – If a positive number is raised to an exponent, no matter how large or small, the result will always be POSITIVE! Example • Graph each of the following. • Describe how each graph can be obtained from the graph of y = ln x. • Give the domain and the vertical asymptote of each function. • a) f(x) = ln (x − 2) • b) f(x) = 2 − ¼ ln x • c) f(x) = |ln (x + 1)| Graph f(x) = 2 − ¼ ln x • The graph is a vertical shrinking by a factor of 14 , followed by a reflection across the x-axis, and then a translation up 2 units. • The domain is the set of all positive real numbers. • The y-axis is the vertical asymptote asymptote. x 0.1 0.5 1 3 5 f(x) 2.576 2.173 2 1.725 1.598 Graph f(x) = ln (x − 2) • The graph is a shift 2 units right. • The domain is the set of all real numbers greater than 2. • The line x = 2 is the vertical asymptote. x f( ) f(x) 2.25 −1.386 2.5 −0.693 3 0 4 0.693 5 1.099 Graph f(x) = |ln (x + 1)| • The graph is a translation 1 unit to the left. Then the absolute value has the effect of reflecting negative outputs across the x-axis. • The domain is the set of all real numbers greater than −1. • The line x = −1 is the vertical asymptote. x −0.5 0 1 3 6 f(x) 0.693 0 0.693 1.386 1.946 5 Application: Walking Speed • In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function w(P) = 0.37 ln P + 0.05. The population of Salem Salem, Oregon is about 137 137, 000 000. Find the average walking speed of people living in Salem. 6