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QUESTION 15 (4 marks) Solve the following equations. a) 4e* =100 [1 mark] b) 2log x—log (x-1)=1 [3 marks]
QUESTION 18 (6 marks) The function f(x) has the form given by f(x) = 3 log2 (x+a)+b The function g(x) has the form given by g(x) =—log (x + ¢) + 5 A section of the graphs of the two functions is shown. X / a oo r Determine the values of a b and c.
QUESTION 2 The pH of a substance is a measure of its acidity and is given by the formula pH = —log; [H ] where [H ]is the concentration of hydrogen ions in moles per litre. If a solution has a pH equal to 0.2 the concentration of hydrogen ions in moles per litre is closest to (A) 0.32 (B) 0.63 (C) 0.70 (D) 1.8
QUESTION 4 Consider the function f(x) =log (x+¢g) wherep>1and 0 <g <. Which of the following could be the graph of f(x)? (A) (B) © (D)
QUESTION 1 2log;o (x)—log;o(3x) is equal to (A) logm[?] (B) log (x2 - 3x) 2logyg (x) log;(3x) (D) —log;o(x) ©)
QUESTION 12 (5 marks) Solve for x in the following. a) log (5x+7)f 5 [2 marks] b) loglu(x‘i‘3)+]0g]0(x_3): logm(gx—29) [3 marks]
QUESTION 14 (4 marks) Consider the function f(x)=In(3x+4) for x > _?4 a) Determine f`(x). [1 mark] b) Determine the x-intercept of the graph of f/(x). [2 marks] c) Determine the gradient of the tangent to the graph of f(x) at the x-intercept. [1 mark]
QUESTION 5 Solve for x given that logy(x—1)=2. (A) 7 (B) 8 © 9 (D) 10
QUESTION 9 The graphs of the functions f(x)=2¢ +5 and g(x)= point A. 3 . : : : — intersect at point A. Determine the coordinates of B A) (1.609 15) (B) (1.099 1) (C) (0.4065 2) (D) (-0.693 6)
QUESTION 15 (4 marks) A new internet search engine gives a ranking R to each website based on the function R = log; (5011r2 ) where / is the number of hits (visits) the website has received. If a website currently has 100 hits determine how many more hits they need to increase their ranking by 1.
QUESTION 6 Which graph represents the function f(x)=-3—In(x+3)? (A) (B) -—+—+—+—+0 —43211012345 (D) Y 5 4 3 2 1 O 0 -— X -4 -3 -2 -1 1 23456 728 910 -4-3-2-1.0 1 2 4 56 78 910 3 -1 B %)) -3 -3 -4 -4 -5 -5 -6 -6 -7 7 -8 -8 —_ N W KA W -1.01 234567238910 © —_— N W KA WD
QUESTION 11 (5 marks) Solve for x in the following. a) ln(2x) =3 [2 marks] b) log (4x+16)log (x* —2)=1 [3 marks]
QUESTION 8 Determine the equation of the asymptote of the function f (x) =logg (x — 3) —4. (A) x=-4 (B) x=-3 (C) x=3 (D) x=4
QUESTION 10 The solution of 2 = 42 is (A) (B) ©) (D) 1.48 231 3.37 4.54
QUESTION 8 Given y = 5e2* rearrange the function to make x the subject. (A) x=In (%) ®) x=j@ (€) x=In(V5y) D) x=In ((%)Z>
QUESTION 12 (3 marks) Simplify the following: a) logg9 + 2logg 2 logz 16 b) log; 64 [1 mark] [2 marks]
QUESTION 13 (4 marks) Solve the following: a) logs(2x+4) =2 [2 marks] b) In(7 —x) —In10 = In(x) [2 marks]
QUESTION 1 Determine the asymptote of the graph of the function f(x) = log;(x +5) (A) x=-5 (B) x=5 ©) y=-5 D) y=5
QUESTION 11 (4 marks) The Palermo Technical Impact Hazard Scale (P) is a logarithmic scale used by astronomers to rate the potential hazard of a Near-Earth Object (NEO). Potential impacts with positive Palermo Scale values will generally indicate situations that merit some level of concern. For the NEO asteroid Apophis P is given by: P = logso (8.71 Zi10—6) where p; represents the impact probability. a) Determine P for a p; value of 1 in 100 000. [1 mark] John believes that increasing the P value in 11a) by 2 will make the chance of impact 1000 times more likely. b) Evaluate the reasonableness of this statement. [3 marks]
QUESTION 20 (5 marks) A farmer has a paddock with straight fences on two sides (fence A and fence B) perpendicular to each other and bounded by a creek on the other side. Fence B is 4 kilometres long. The creek boundary can be modelled using d=In(5x+¢e)—1 where d is the perpendicular distance in kilometres from fence A to the creek and x is the distance in kilometres along fence A from the point O. The farmer wants to divide his paddock area in half with a straight fence parallel to fence B. Not drawn to scale creek Fence B O Fence A Determine where the farmer should locate the fence.
QUESTION 16 (5 marks) Solve for x in the equation 4+ 7e?* = 3e”.
QUESTION 19 (7 marks) Jaxon and Shari each own a shop and have recorded the number of customers entering their shop on two consecutive days. The number of daily customers for each shop can be modelled by the equation y = A In( Bx) where x is the day and y is the number of customers. The constants A and B are different for each shop. Determine algebraically whether the total number of customers for Jaxon and Shari’s shops will be the same every day in the future.
QUESTION 5 Solve In(x) + In(3.70) = In(9.25) for x. (A) 0.92 (B) 1.71 (C) 2.50 (D) 5.55
QUESTION 15 (4 marks) Determine the derivative of f (x) =Inx? + In(x —35 y. Express the derivative as a single fraction in its simplest and factorised form.
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