Express y as a function of x. C is a positive number.Try These (6.4-6.5)Sullivan , 8th ed. MAC1140/MAC11476-4: Logarithmic FunctionsThe inverse function of y = f(x) = ax is x = ay, which is equivalent to the logarithmic function y = log a x. This equivalence is used to convert between logarithmic and exponential functions. If the base is the number e, then x = e y if and only if y = ln x.Write the equation in logarithmic form: Write in exponential form:1. 16 = 422. e2.2 = M 3. logb 4 = 2 4. ln x = 4Evaluate; if au = av, then u = v: 7. Find a so that the graph of f(x) = logax.5. log5 2536. ln e3contains the point (32, 8); simplify.y = loga xor x = ayDomain Range x-intercept VerticalasymptoteCharac-teristicPasses througha > 1 [0, ) (-, ) (1, 0) y-axis as y -Increasing,one-to-one(1, 0) (a, 1)0 < a < 1 [0, ) (-, ) (1, 0) y-axis as yDecreasingone-to-one(1, 0) (a, 1)Graph the given logarithmic function and its inverse; state the domain, range, and any vertical asymptote:8. f(x) = 3 – ln (x + 2) 9. f(x) = 2 + log1/2 xx  e3-y - 2e(3 - x) - 2= y x = (1/2)y-2(1/2)x-2= y0 -11 02 13 24 3Sullivan , 8th ed. MAC1140/MAC11476.5: Properties of LogarithmsProperties: (1) alog 1 0 and alog a 1; (2) alog Ma M and ralog a r;(3)  a a alog MN log M log N ; (4) a a aMlog log M log NN ; (5) ra alog M r log M; (6) a alog log NN    1. If M, N, and a are positive real numbers and a 0 and b0, then (7) if M = N, a alog M log N, and (8) if a alog M log N ,M = N. Change of base formulas: (9) bablog Mlog Mlog a and (10) aln Mlog Mln a.Use the properties of logarithms to find the exact value of each expression. Do not use a calculator:1.ln e22.log log6 69 43.+log log7 715 37Use properties of logarithms to rewrite the logarithm in terms of p, q r, or s:4. If ln 4 = p, 5. If ln 10 = q and ln 18 = r, 6. If ln5 = r and ln 225 = s,ln 256 = ln1.8 = ln345= Write as the sum or difference of logarithms: 7.bx ylogz5 2368.     xlnx223241Express as a single logarithm:9. b b blog x log x log y 2 310.( )log log logx x+ -74 4 435 2 2 5 3 3Evaluate; round your answer to nearest hundredth.Sullivan , 8th ed. MAC1140/MAC114711. log5 18 12. log 2p 13.log5 343 . log725 14. log24 . log4 6 . log6 8Express y as a function of x. C is a positive number.15. ln y = 15x + lnC 16. 5ln(y) = ( )ln( ) ln lnx x C- - + +1 15 73 5Sullivan , 8th ed. MAC1140/MAC1147Try These (6.4-6.5)Use the properties of logarithms to find the exact value of each expression. Do not use a calculator:1. Graph the function: y = -log(x – 3) + 2Domain: {x  x > 3} Range: (-  ,  ) Asymptotes: x = 3 2. a. log816 – log82 b.e xln 29log88 = 1 x = 29Write as the sum or difference of logarithms:3.( )= + - --++3213log log( 1)1log 2log 2)22(x xxx x xExpress as a single logarithm:4.( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )� �+ -+= �� �+ - + +� �� � � �+ - + -= � =� �+ - + +-- +� �- + + - + +� � � �2 222 3 7 6log l3 12log2 2 6 13 1 3 11log log2 6 1 2 6 1og4 2x xxx x x xx x x xx x x x x xx x x xx xExpress y as a function of x. The constant C is a positive number.5. log(y + 8) = 3x + log Clog (y + 8) – log C = 3x38log 3810xyxCyC+=+= y + 8 = 103xCy = 103xC -