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Accelerated Integrated Geometry 7.1 Logarithmic Functions * Logarithms: used to find unknown exponents in exponential form Exponential Form Logarithmic Form bY =x y =log,x Example: 10° = 1000 3=10g,)1000 e Examples: Fill in the blanks below. Exponential Form Logarithmic Form 3 (3) =64 log o4=-3 4 4e 3*= 8 ——>» log,81= 4 ipooe 1 ]@ 2 <> log,,4 = 3 e Common Logarithm: logarithm with base 10 e Examples: Solve for x. 3, 10*=85 4. 10°: 109 ; X= 1091 g5 X= logic Tot x= 1.929 = 2,037 ¢ Exponential and Logarithmic functions as inverses: Exponential: y = 10” Logarithmic: y =log,ox > | ‘|y=x i Graphically: iim K How can we prove these two equations are inverses algebraically? By compositions « Examples: Find the value of a. 5. 4=log,a 6. a=log,64 7. a=logy55 stra 4*= 64 5*= 5 Blza ues 4? Stas § a3 3a =| ot a3 8. 5=log,32 9. 1=log,a 10. a=log,1 a’ = 32 B=a 10% = | S- are 2° Z=a a=0 =k 11. a=log,343 12, 5 loge 13, log,243=5 7°= 343 Bieae a®= 243 a 3 =O. Ss 5 Te F ee gt a= 3 a=3 gra a=3 « Analyzing Graphs of Logarithmic Functions: Graph of y=log,x, a>1 in: Increasing/ . ( Domain: (0, oo) Decreasing: Increasing 0, oo) Range: (-co J eo) Asymptote: Nerical esymapiote Intercept: “ ne Continuous?: Yes « Examples: Evaluate. 7. log,8 8. log,16 272 § Y*= 16 x23 x= 2 9. logg25* (referved to inverse properties) 10, 10g,27'° Cinverse) y logs 25 100 logs aq y log e252 100 oD 3 4. zB 100+ 32 300 11, 384 +109,25 (inverse 12. log,32-5°%? Cinuers) Orr at s_ 4+ logs loge® 3 442 5-3 @ 2 ° Examples: Solve for x. Then, check your solutions. 13. log, (x? +7x-5) =log, (6x +1) (one-to-one) 2 - * erp aon pa logs €17) 2undefined Seer igs (8) - (2-2) (243920 X-2=0 x+3=0 Zea a4=-3 14, log, (2x? +8x-11) =log, (2x+9) 2 - ** Bal a gl LZ5 loga (-!) > undefined 22?+6x-20= 0 log 2 (13) @a-4) (x+5)= 6 Mahe Se eT EHS 15. log, (x2) Tog, =log, (x+2) loge CS) = loge (x#2) a-% Accelerated Integrated Geometry 7.3 The Natural Base, € Natural Exponential Function: TOY f(x) =e* Graph: _| natural base ) Examples: Evaluate f(x) =e* with the following values of x. {fj | 1 1, x=2 : 2 KES 3. x=-l 2 o as ' 1 eF= (2.718)"= 7,384] i seme eF= (18) |, 6487 Fe za 0.3679 Continuous Compounding Formula: a, interest - amount+— A =Pe™ \ Principal —_) time in years (initial) Examples: 4. An investment of $1000 earns an annual interest rate of 7.6%. If the interest is compounded continuously for 8 years, find the final amount. P= 1,000 Az 1,000 (2.718) 02%") = . O16 48 yoo Az 5) 836.75 5. $2000 is invested at an annual interest rate of 9%. If the interest is compounded continuously for 12 years, find the final amount. P= £2,000 Az 2,000 (2,718) 0") R= 0.09 _¢ i yo A= $5,884.36 Accelerated Integrated Geometry 7.4 Solving Exponential and Logarithmic Equations + Inverse Properties: Forb>Oandb 1... log,b* = x bee* = x One-to-One Property: If b* =b’, thenx=y. ¢ One-to-One Property: Tf log,x =logy, then x = y. * Change-of-Base Formula: [roma * Examples: Solve each exponential equation or inequality algebraically. 1 7% = 73 2. 125* = 25" X=2x-3 53 _ 5re+d Thx “2x 5 Beate “ees 3x = Ze+2 x= 3 2m 72% haar, x3 3. 8% = 8%" 4, = 5) 5x 3x44 \r2aee /1\x-3 73x _~3x G6) Pd G 2224 >> rane x-3 2 2 Ax es LEZ ‘ -3x2-3 xa=1 5. 4e%? =72 6. 62° +8=20 as tos exS= 1g Ger**= 0 % z loge!8 = 3x-5 err. 7 0.25% = 0.693 ore wraraie (Soe Os +5.00 45 dn 2= 0.25% X= 2.71726 3x = 7.89 0,C93= 0, 25% X= 2.63 _ enn LOGARITHMIC « Examples: Evaluate. 7. log,56=x 8. log,97 =x log 5G Jog 7 i] log 8 2.0686 = x 2.202% + Examples: Solve each equation for x. 9. log, 7x =log, (x? +12) 10. log(x+48)+logx =2 Tn = 22412 log (xCe+48)) = 2 “Ix Tx logjolx?+ Y8x)= 2 Oz xt Tx 12 OP = x24 4Bx Oz be Da-Y “100-100 x=3 x24 xt+48x-IDO=O GtSOG-2)= 0 x#-50 x22 11, logx +log(x-3)=1 12, 2Inx+2=1 10,(x(x-3)) = L 4% 2 Rinxe-1 SOte x -3% Iz FT 10-10 tn xne-¥ 2-3x-1020 a G29) 20 lngex =~ x¢-2 x25 xe? x= 0.6065 13, In(7x -13) =In(2x +17) Tx-IB = 24417 “2ntss “Ax tl3 Sx+ 30 Ss 5 x= © GRAPHICALLY Solving Inequalities Using a Graph: 1, Graph each side of inequality separately 2. Find the intersection (x,y) 3. Find where x-value of first graph is less than or greater than the second graph * Examples: Use a graphing calculator to solve the following inequalities. 7. BP ch 8. log, x > A _ C0644) t reves Ley] > 1 | aut i" Domdin ter His Corner Coe 71.060) gPomain = (1.1331, 00) 113311) 
