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LogarithmsQuestion and Answers: Page 6

Question Number 167176    Answers: 0   Comments: 1

Question Number 166687    Answers: 1   Comments: 1

log _((2x−1)) (x+1) > log _((4−2x)) (x+1) x=?

$$\:\:\:\mathrm{log}\:_{\left(\mathrm{2x}−\mathrm{1}\right)} \left(\mathrm{x}+\mathrm{1}\right)\:>\:\mathrm{log}\:_{\left(\mathrm{4}−\mathrm{2x}\right)} \left(\mathrm{x}+\mathrm{1}\right) \\ $$$$\:\:\mathrm{x}=? \\ $$

Question Number 165818    Answers: 1   Comments: 0

A uniform sphere of weight W rest between a smooth vertical plane and a smooth plane inclined at an angle θ with the vertical plane. Find the reaction at the contact surfaces.

$$\mathrm{A}\:\mathrm{uniform}\:\mathrm{sphere}\:\mathrm{of}\:\mathrm{weight}\:{W} \\ $$$$\mathrm{rest}\:\mathrm{between}\:\mathrm{a}\:\mathrm{smooth}\:\:\mathrm{vertical} \\ $$$$\mathrm{plane}\:\mathrm{and}\:\mathrm{a}\:\mathrm{smooth}\:\mathrm{plane}\:\mathrm{inclined} \\ $$$$\mathrm{at}\:\mathrm{an}\:\mathrm{angle}\:\theta\:\mathrm{with}\:\mathrm{the}\:\mathrm{vertical} \\ $$$$\mathrm{plane}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{reaction}\:\mathrm{at}\:\mathrm{the}\: \\ $$$$\mathrm{contact}\:\mathrm{surfaces}.\: \\ $$

Question Number 165178    Answers: 2   Comments: 0

x∈R ⇒ ∣ log _2 ((x/2))∣^3 +∣log _2 (2x)∣^3 =28

$$\:\:{x}\in{R}\:\Rightarrow\:\mid\:\mathrm{log}\:_{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\mid^{\mathrm{3}} +\mid\mathrm{log}\:_{\mathrm{2}} \left(\mathrm{2}{x}\right)\mid^{\mathrm{3}} =\mathrm{28} \\ $$

Question Number 165136    Answers: 3   Comments: 0

Question Number 165068    Answers: 0   Comments: 1

Question Number 164985    Answers: 2   Comments: 0

Question Number 163061    Answers: 1   Comments: 0

(2)^(1/(log _x (((243)/x)))) = ((log _x ((x^5 /9))))^(1/3)

$$\:\sqrt[{\mathrm{log}\:_{{x}} \left(\frac{\mathrm{243}}{{x}}\right)}]{\mathrm{2}}\:=\:\sqrt[{\mathrm{3}}]{\mathrm{log}\:_{{x}} \left(\frac{{x}^{\mathrm{5}} }{\mathrm{9}}\right)}\: \\ $$

Question Number 163060    Answers: 1   Comments: 1

2log _3 ((x^2 /(27))) = 2+ ((log _3 ((1/x)))/(log _5 ((√x) )))

$$\:\:\mathrm{2log}\:_{\mathrm{3}} \left(\frac{{x}^{\mathrm{2}} }{\mathrm{27}}\right)\:=\:\mathrm{2}+\:\frac{\mathrm{log}\:_{\mathrm{3}} \left(\frac{\mathrm{1}}{{x}}\right)}{\mathrm{log}\:_{\mathrm{5}} \left(\sqrt{{x}}\:\right)}\: \\ $$

Question Number 164806    Answers: 1   Comments: 7

Question Number 161786    Answers: 2   Comments: 0

logx_(4x) +logx_(x/2) =2 solve for x=?

$${log}\underset{\mathrm{4}{x}} {{x}}+{log}\underset{\frac{{x}}{\mathrm{2}}} {{x}}=\mathrm{2} \\ $$$${solve}\:\:\:{for}\:\:\:{x}=? \\ $$

Question Number 161444    Answers: 0   Comments: 2

Question Number 161362    Answers: 0   Comments: 1

log _(√(x/3)) (3x−54)^(log _3 (x)) = 18−3log _(x/3) (x^2 ) x=?

$$\:\mathrm{log}\:_{\sqrt{\frac{{x}}{\mathrm{3}}}} \left(\mathrm{3}{x}−\mathrm{54}\right)^{\mathrm{log}\:_{\mathrm{3}} \left({x}\right)} \:=\:\mathrm{18}−\mathrm{3log}\:_{\frac{{x}}{\mathrm{3}}} \left({x}^{\mathrm{2}} \right) \\ $$$$\:{x}=? \\ $$

Question Number 161342    Answers: 2   Comments: 0

2log _x (3) log _(3x) (3)=log _(9(√x)) (3) x=?

$$\:\:\mathrm{2log}\:_{\mathrm{x}} \left(\mathrm{3}\right)\:\mathrm{log}\:_{\mathrm{3x}} \left(\mathrm{3}\right)=\mathrm{log}\:_{\mathrm{9}\sqrt{\mathrm{x}}} \left(\mathrm{3}\right) \\ $$$$\:\mathrm{x}=? \\ $$

Question Number 160987    Answers: 1   Comments: 0

Solve for x log _(log _6 (x−1)) (64) = 6

$$\:\mathrm{Solve}\:\mathrm{for}\:\mathrm{x}\: \\ $$$$\:\:\:\:\:\:\mathrm{log}\:_{\mathrm{log}\:_{\mathrm{6}} \left(\mathrm{x}−\mathrm{1}\right)} \left(\mathrm{64}\right)\:=\:\mathrm{6}\: \\ $$

Question Number 160931    Answers: 1   Comments: 0

x = 2^(log _5 (x+3)) ; x=?

$$\:\:{x}\:=\:\mathrm{2}^{\mathrm{log}\:_{\mathrm{5}} \left({x}+\mathrm{3}\right)} \:;\:{x}=? \\ $$

Question Number 160426    Answers: 0   Comments: 2

Question Number 159762    Answers: 1   Comments: 0

Given log _3 (n)= log _6 (m)=log _(12) (m+n) (m/n) = ?

$$\:\:\:{Given}\:\mathrm{log}\:_{\mathrm{3}} \left({n}\right)=\:\mathrm{log}\:_{\mathrm{6}} \left({m}\right)=\mathrm{log}\:_{\mathrm{12}} \left({m}+{n}\right) \\ $$$$\:\:\:\frac{{m}}{{n}}\:=\:? \\ $$

Question Number 159639    Answers: 1   Comments: 0

((x−1)/(log _3 (9−3^x )−3)) ≤ 1

$$\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{x}−\mathrm{1}}{\mathrm{log}\:_{\mathrm{3}} \left(\mathrm{9}−\mathrm{3}^{\mathrm{x}} \right)−\mathrm{3}}\:\leqslant\:\mathrm{1}\: \\ $$

Question Number 159507    Answers: 2   Comments: 1

Question Number 159157    Answers: 1   Comments: 0

Question Number 159122    Answers: 0   Comments: 0

In order to monitor buses in a travel agency, the manager decides to monitor the number of break downs of the buses using the sequence {x_n } defined by x_(n+1) = 1.05 x_n + 4. Given that x_0 = 40. is the number of break downs by the buses from the 1^(st) of january 2000, and that for every n∈N, we denote x_n the number of breakdowns of the buses as from 1^(st) of january of the year (2000 + n) (a) Calculate x_1 , x_2 , x_3 (b) Consider the sequence {y_n } defined by y_n = x_n + 80 for all n ∈ N (i) express y_(n+1) in terms of y_n and deduce the nature of the sequence {y_n }. (ii) Express y_n in terms of n. deduce x_n in terms of n (iv) find the number of break downs that will be registered by 1^(st) january 2021.

$$\mathrm{In}\:\mathrm{order}\:\mathrm{to}\:\mathrm{monitor}\:\mathrm{buses}\:\mathrm{in}\:\mathrm{a}\:\mathrm{travel} \\ $$$$\mathrm{agency},\:\mathrm{the}\:\mathrm{manager}\:\mathrm{decides}\:\mathrm{to}\:\mathrm{monitor} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{using}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{x}_{{n}} \right\}\:\mathrm{defined}\:\mathrm{by} \\ $$$${x}_{{n}+\mathrm{1}} \:=\:\mathrm{1}.\mathrm{05}\:{x}_{{n}} \:+\:\mathrm{4}.\:\mathrm{Given}\:\mathrm{that}\:{x}_{\mathrm{0}} \:=\:\mathrm{40}. \\ $$$$\mathrm{is}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs}\:\mathrm{by}\:\mathrm{the}\:\mathrm{buses} \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{of}\:\mathrm{january}\:\mathrm{2000},\:\mathrm{and}\:\mathrm{that} \\ $$$$\mathrm{for}\:\mathrm{every}\:{n}\in\mathbb{N},\:\mathrm{we}\:\mathrm{denote}\:{x}_{{n}} \:\mathrm{the}\:\mathrm{number} \\ $$$$\mathrm{of}\:\mathrm{breakdowns}\:\mathrm{of}\:\mathrm{the}\:\mathrm{buses}\:\mathrm{as}\:\mathrm{from}\:\mathrm{1}^{\mathrm{st}} \\ $$$$\mathrm{of}\:\mathrm{january}\:\mathrm{of}\:\mathrm{the}\:\mathrm{year}\:\left(\mathrm{2000}\:+\:{n}\right) \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Calculate}\:{x}_{\mathrm{1}} ,\:{x}_{\mathrm{2}} \:,\:{x}_{\mathrm{3}} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}\:\mathrm{defined} \\ $$$$\mathrm{by}\:{y}_{{n}} \:=\:{x}_{{n}} \:+\:\mathrm{80}\:\mathrm{for}\:\mathrm{all}\:{n}\:\in\:\mathbb{N} \\ $$$$\left(\mathrm{i}\right)\:\mathrm{express}\:{y}_{{n}+\mathrm{1}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{y}_{{n}} \:\mathrm{and} \\ $$$$\mathrm{deduce}\:\mathrm{the}\:\mathrm{nature}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence}\:\left\{{y}_{{n}} \right\}. \\ $$$$\left(\mathrm{ii}\right)\:\mathrm{Express}\:{y}_{{n}} \:\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n}.\:\mathrm{deduce}\:{x}_{{n}} \\ $$$$\mathrm{in}\:\mathrm{terms}\:\mathrm{of}\:{n} \\ $$$$\left(\mathrm{iv}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{number}\:\mathrm{of}\:\mathrm{break}\:\mathrm{downs} \\ $$$$\mathrm{that}\:\mathrm{will}\:\mathrm{be}\:\mathrm{registered}\:\mathrm{by}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{january}\: \\ $$$$\mathrm{2021}. \\ $$

Question Number 156109    Answers: 2   Comments: 0

log _5 ((√(x−9)))−log _5 (3x^2 −12)−log _5 ((√(2x−1))) ≤ 0

$$\:\:\mathrm{log}\:_{\mathrm{5}} \left(\sqrt{\mathrm{x}−\mathrm{9}}\right)−\mathrm{log}\:_{\mathrm{5}} \left(\mathrm{3x}^{\mathrm{2}} −\mathrm{12}\right)−\mathrm{log}\:_{\mathrm{5}} \left(\sqrt{\mathrm{2x}−\mathrm{1}}\right)\:\leqslant\:\mathrm{0} \\ $$

Question Number 155992    Answers: 0   Comments: 0

(1+log _3 x).(√(log _(3x) ((x/3))^(1/3) )) ≤ 2

$$\:\left(\mathrm{1}+\mathrm{log}\:_{\mathrm{3}} \:\mathrm{x}\right).\sqrt{\mathrm{log}\:_{\mathrm{3x}} \:\sqrt[{\mathrm{3}}]{\frac{\mathrm{x}}{\mathrm{3}}}}\:\leqslant\:\mathrm{2} \\ $$

Question Number 155638    Answers: 1   Comments: 2

Question Number 155583    Answers: 2   Comments: 0

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