Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from grph or in problem sttement (F IF.2) recognizing tht logrithmic function is the inverse of the relted exponentil function by compring their grphs on the sme xes (F IF.5) Introduction In this course, you hve studied vriety of functions, such s trigonometric functions, qudrtic functions, nd the inverses of functions. You hve worked with exponents in the pst nd probbly relize tht exponents re not lwys whole numbers. You my lso recll tht sometimes exponents contin vribles. An exponentil function is function tht hs vrible in the exponent, such s f(x) = 5 x. The power is the result of rising bse to n exponent; 32 is power of 2 since 2 5 = 32. The exponent is the vlue of the function s corresponding logrithm, such s x in the logrithmic function x = log 5 f(x) nd its corresponding exponentil function, f(x) = 5 x. Like other functions, exponentil functions hve inverses, which re clled logrithmic functions. A logrithmic function is the inverse of n exponentil function. For exmple, for the exponentil function f(x) = 5 x, the inverse logrithmic function is x = log 5 f(x). If the exponentil function is of the form f(x) = x, then the logrithmic function is of the form log f(x) = x. This confirms the reltionship between function f(x) = y nd its inverse, g(y) = x. This reltionship cn lso be seen from the following grph of n exponentil function, f(x) = 10 x, nd its inverse logrithmic function, log 10 f(x). U1-95
Lesson : Logrithmic Functions s Inverses 10 y 8 f(x) = 10 x 6 4 y = x 2 10 8 6 4 2 0 2 4 6 8 10 2 log 10 f(x) x 4 6 8 10 Notice tht the exponentil function nd its inverse logrithmic function re reflected cross the line f(x) = x (often written s y = x). For exmple, this mens tht for the vlue x = 3, the exponentil function is given by f(3) = 10 3 nd its inverse logrithmic function is log 10 f(3) = log 10 (10 3 ) = 3. In rel-world problems, such s the sound-intensity exmple in the Wrm-Up, there will be situtions in which the inverse function is more effectively used thn the function from which the inverse is derived. Knowledge of the rel-world domin of the function cn help mke the decision bout whether the function or its inverse hs more mening. Another fctor in deciding which function to work with is how simplified the expressions nd numbers re for ech function. Key Concepts As the grph in the Introduction shows, the exponentil function nd its inverse re one-to-one over their domins. The domin of the exponentil function is (, + ). However, the domin of the logrithmic function is (0, + ). The rnge of the exponentil function is (0, + ). The rnge of the logrithmic function is (, + ). This informtion provides more evidence tht the logrithmic function is the inverse of the exponentil function. U1-96
Lesson : Logrithmic Functions s Inverses In the grphed exmple, the vlue of the exponentil function is 1 t x = 0 becuse f(0) = 10 0 = 1. Correspondingly, the vlue of the inverse logrithmic function is 0 t x = 1 becuse log 10 (1) = 0. Exponentil functions with more constnts cn be explored using the properties of exponents or by looking t dt tbles generted by grphing clcultor. Use grphing clcultor to explore the domin, rnge, nd other key points of the function 4 3 2x nd its inverse logrithmic function by looking t dt tbles of domin nd function vlues. Follow the directions pproprite to your clcultor model. On TI-83/84: Step 1: Press [Y=]. Press [CLEAR] to delete ny other functions stored on the screen. Step 2: At Y1, use your keypd to enter vlues for the function. Use [X, T, θ, n] for x nd [x 2 ] for ny exponents. Step 3: Press [GRAPH]. Press [WINDOW] to djust the grph s xes. Step 4: Press [2ND][GRAPH] to disply tble of vlues. Look t the domin vlues round x = 0. On TI-Nspire: Step 1: Press [home] to disply the Home screen. Step 2: Arrow down to the grphing icon, the second icon from the left, nd press [enter]. Step 3: Enter the function to the right of f1(x) = nd press [enter]. Step 4: To djust the x- nd y-xis scles on the window, press [menu] nd select 4: Window nd then 1: Window Settings. Enter ech setting s needed. Tb to OK nd press [enter]. Step 5: To see tble of vlues, press [menu] nd scroll down to 2: View, then 5: Show Tble. Either clcultor will show exponentil function vlues tht pproch 0 s x becomes negtive nd tht increse s x becomes positive. U1-97
Lesson : Logrithmic Functions s Inverses To show the corresponding function vlues for the inverse logrithmic function, switch the x- nd y-vlues, s shown in the following tble. Exponentil function Logrithmic function x 2 1 0 1 2 y 0.05 0.44 4 36 324 x 0.05 0.44 4 36 324 y 2 1 0 1 2 Notice tht the logrithmic function does not exist for negtive domin vlues. The logrithmic function vlues cn be verified with the dt tble. 1 f x For exmple, f(x) = 4 3 2x, so x = 2 log ( ) f (0) 3. For x = 0, log3 4 = 0 4 4 log 4 log 3 (1) = 0. 3 = Notice tht the coefficient of 4 in the function chnges the vlue of the function to 4 t x = 0, nd it chnges the vlue of x to 4 when the vlue of the inverse function is 0. Finlly, the bsic definitions nd rules of exponents nd logrithms will be needed in order to mnipulte nd clculte exponentil nd logrithmic functions, summrized s follows. Terms nd Rules for Logrithms In logrithmic eqution, log b = c, is the bse, b is the rgument, nd c is the logrithm of b to the bse. The bse is the quntity tht is being rised to n exponent in n exponentil expression, such s in the expression x, or the quntity tht is rised to n exponent which is the vlue of the logrithm, such s 2 in the function log 2 g(x) = 3 x. The rgument is the result of rising the bse of logrithm to the power tht is the vlue of the logrithm, so tht b is the rgument of the logrithm log b = c. U1-98
Lesson : Logrithmic Functions s Inverses You my recll the rules for working with exponents; for exmple, ccording to the Product of Powers Property, when multiplying two exponents with the sme bse, keep the bse nd dd the powers: x y = x + y. The rules for vrious opertions with logrithms re derived from the rules for exponents. The following tble lists some exponent rules, followed by the eqution nd nme of the relted logrithmic rule. Exponent rule Relted logrithm rule Logrithm rule nme x y = x + y log (x y) = log x + log y Product rule x y x x y = log x y y = log log Quotient rule ( x ) y = x y log x y = y log x Power rule Another rule, the bse chnge rule, llows for computing with logrithms other thn bse 10; log10 one form of the eqution for this rule is log b =. (Other forms will be discussed lter.) log b This rule is prticulrly useful when working with clcultors tht only clculte with logrithms with bses of e (nturl logrithms) nd 10 (common logrithms). The irrtionl number e hs vlue of pproximtely 2.71828. A nturl logrithm is logrithm with bse of e. Nturl logrithms re usully written in the form ln, which mens log e. For exmple, f(x) = ln (1 x) is understood to be the inverse of the function for the exponentil function g(x) = 1 e x. A common logrithm, on the other hnd, is logrithm with bse of 10. When writing common logrithm, the 10 is usully omitted, such tht log x = log 10 x. For exmple, the logrithmic function f(x) = log x is understood to be the inverse function for the exponentil function g(x) = 10 x. 10 U1-99
Lesson : Logrithmic Functions s Inverses Common Errors/Misconceptions incorrectly identifying the domin nd rnge vribles in n exponentil function nd in its inverse logrithmic function confusing the bse with the power in expressing n exponentil function s logrithmic function, or vice vers misidentifying the domins of exponentil functions nd their inverse logrithmic functions misinterpreting the coefficients of bse nd of vrible in power in n exponentil function when writing the inverse logrithmic function mispplying the rules of exponents nd logrithms in rewriting exponentil nd logrithmic functions U1-100