Name: Date: Block: Exponents/Logs and Sequences and Series TEST STUDY GUIDE

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1 Algebra STUDY GUIDE AII.6, AII.7, AII. Exponents/Logs, Sequences and Series Mrs. Grieser Name: Date: Block: Test covers: Exponents/Logs and Sequences and Series TEST STUDY GUIDE Sketch graphs of exponential and logarithmic functions, finding domain, range, and asymptotes. Use transformations to graph exponential and logarithmic functions. Find inverses of exponential and logarithmic functions Expand logarithms using the properties of logs. Condense logarithms using the properties of logs. Use the change of base formula to estimate values of logarithms using the calculator. Solve exponential and logarithmic equations, rejecting extraneous solutions. Know the difference between sequences and series. Identify whether a sequence is arithmetic, geometric, or neither. Find explicit and recursive rules for arithmetic and geometric sequences. Find n th terms of sequences. Find missing arithmetic or geometric means for a given sequence. Evaluate series using summation notation. Evaluate partial sums of arithmetic or geometric series. Identify whether a geometric series will converge or diverge. Find the sum of an infinite geometric series. BRING YOUR SOL FORMULA SHEETS! You may use them on the test. Practice Questions: ) Graph the functions. State domain, range, and asymptotes. Use table of values and transformations as appropriate. a) Graph y = x b) Graph y = logx

2 Algebra STUDY GUIDE AII.6, AII.7, AII. Exps/Logs, Sequences and Series Mrs. Grieser Page c) Graph y = x+ d) Graph y = log(x-) e) Graph y = x - + f) Graph y = log(x+) - 4 ) Find the inverses of the functions. a) y e 4 x b) y = x c) y = ln x + 8 d) y = log6(x+0) ) Expand the logarithms. a) log xy b) ln xy 4 x zx c) log d) ln y y 4

3 Algebra STUDY GUIDE AII.6, AII.7, AII. Exps/Logs, Sequences and Series Mrs. Grieser Page 4) Condense into one logarithm. a) log4 + log4 b) 9 log6 + log c) 6 ln 7 + ln6 d) 4logx + 5 logz = 8 logy 5) Estimate the value of the logs using the change of base formula to the nearest thousandth. Show the change of base formula you used. a) log7 b) log000 c) log 57 d) log65 6) Solve the equations. Check for extraneous solution. a) -4-6x 0 = -8 b) e 6-x + 7 = 8 c) 7e -p = -7 d) 7 4x- = 9 x+8 e) log7(-n-4) = log7(-n-0) f) log4(n-) = log4(5n-4) g) -7 log4(n +0)=-8 h) log x log (x+) = i) log(x) + log(x-) = j) log9(x+0)+log9x=log99 k) log(x+)(x+4) - log(x+)4 = 7) Determine whether the sequence is arithmetic, geometric, or neither. If it is arithmetic, find the common difference d; if it is geometric, find the common ratio r. a), 5, 9, 4 b) 00, 50,.5,.6, c), 0, 7, 4, d) -, -, -/, -/4 8) Write both an explicit and recursive rule for each sequence. Then find the 8 th term. a) 8, 8, 8, b) a9 = 6, d = 6 (arithmetic sequence) c)a0=0 and a7=48 (arithmetic sequence) d) a=-50 and a6=650 (geometric sequence)

4 Algebra STUDY GUIDE AII.6, AII.7, AII. Exps/Logs, Sequences and Series Mrs. Grieser Page 4 9) Find the first terms of the sequence defined by an = 7 + (n - )5. 0) Find the missing means for the sequences given. a) Find the three arithmetic means between - and -0 b) Find the four arithmetic means between 9 and 54 c) Find the three geometric means between - and -768 d) Find the four geometric means between - and - ) If the fifth and eighth terms of an arithmetic sequence are -9 and -, respectively, what are the first four terms of the sequence? ) Evaluate the partial sums of the series. 0 a) (4k 0k ) b) n ( n ) c) (7k 5) d) k 7 n k n n ) Find the partial sums of the arithmetic or geometric series. a) , n = b) a =, a =85 (arithmetic) c) a =, d=4, n=5 (arithmetic) d) , n = 6 e) a =4, a 9 =04, r=- (geometric) f) a =, a n =-04, r=- (geometric) g) (n ) h) (6i ) n 0 i 8 i) 5 k k 9 j).5 k k

5 Algebra STUDY GUIDE AII.6, AII.7, AII. Exps/Logs, Sequences and Series Mrs. Grieser Page 5 4) Determine the number of terms (n) in each series. a) a =9, a n =9, S n =50 (arithmetic) b) a =4, r=-, S n =-0 (geometric) n c) ( i 4) 80 d) i n k k 40 5) Determine whether the infinite series will converge or diverge. Explain your answer a)... b) ) Find the sum of the infinite series. a) n 4 n b) k 4( 5) k c) n 5 n d) 4.(0.8) k k 7) Driving a piling into a harbor bottom, a pile driver sinks the piling 4 inches on the first stroke, 8 inches on the second stroke, and and / inches on the third stroke. If the sequence is continued, how far will the piling be driven down on the 5th stroke? 8) An outdoor theater has 7 seats in the first row, 40 seats in the second row, and 4 seats in the third row. If this pattern continues, what is the total number of seats in the first 0 rows? 9) A pendulum that is released to swing freely travels 0 inches on the first swing. On each successive swing, the pendulum travels 75% as far as the previous swing. What it the total distance the pendulum swings?

6 Algebra STUDY GUIDE AII.6, AII.7, AII. Exps/Logs, Sequences and Series Mrs. Grieser Page 6 STUDY GUIDE ANSWERS ) a) D={x x} R={y y>0} y = 0 b) D={x x>0} R={y y} x = 0 c) D={x x} R={y y>0} y = 0 d) D={x x>} R={y y} x = e) D={x x} R={y y>} y = f) D={x x>-} R={y y} x = - ) a) y = ln(4x-) b) y = logx c)y = e x-8 d) y = 6 x - 0 ) a) log x log y b) ln x + ln y c) 4 log 9 x 6 log 9 y d) ln z + 4 ln x 6 ln y 4 5 x z 4) a) b) log (6 9 ) c) ln( 7 6 6) d) 8 y 5) a) ln(7)/ln().579 b) ln(00)/ln(0).57 c) ln(57)/(ln(/)) d)ln(5)/ln(6).6 (note: can use log 0 instead of ln) 6 ln 7 9 6) a)x= b) x= c) no solution d)x = e) no solution f) g) x=46 h) no solution i) x= (reject -) 6 j) x= (reject -) k) x=- (reject ) 7) a) neither b) neither c) arithmetic, d = 7 d) geometric r = 8) a) a n =48-0n; a n = a n- -0; a 8 =- b) a n =+6n; a n = a n- +6; a 8 =50 c) a n =-0+4n; a n = a n- + 4, a 8 = d) a n =-(-5) n- ; a =-, a n = (-5)a n- ; a 8 =5650 9) 7,, 7 0) a) -4, -6, -8 b) 4, 9, 44, 49 c) -, -48, -9 d) -, -4, -8, -6 ) 7,, -, -5 ) a) 060 b) 4 c) 57 d) 8 ) a) 0 b) 67 c) 750 d) 78 e) 684 f) -68 g) 99 h) 460 i) 95 j) ) a) n=0 b) n=4 c) 5 d) n=4 5) a) converges, because r=/, which is less than b) diverges because r=-.5 has magnitude > 6) a) 8 b) diverges c) 5 d) 7) inches (4/ in) 8) 505 seats 9) 0 inches