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Университет имени Сулеймана Демиреля

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CHAPTER REVIEW TEST 3G

1. Solve log3(log8 x)	= 0 for x.
A) {3} B) {4}	C) {5} D) {8} E) {16}

2. Solve 52x+1 = 1251 for x.

A) {3} B) {2} C) {1} D) {–1} E) {–2}

3. What is the solution set for 4x – (9		2x) + 8 = 0?
A) {0, 2}	B) {0, 3}	C) {2, 3}
D) {3, 4}	E) {1, 8}

4. Solve log(2x + 1) – log x = 2.

A)		1	B)		1	C)		1	D)	1	E)	1
	104			102			100			99		98

5. What is the solution of the inequality log3(4 – x) < 1?

A) x > 1	B) x > 4	C) x > –1
D) x (1, 4)	E) x (–1, 4)

6. What is the sum of the solutions of log27 a log9 a = 241 ?

A) ñ3 B)	4 3	C)	5 3	D) 2ñ3 E)	7 3
	3		3		3

7.log3 a2 + log2 b3 = 12 and log3 a3 – log2 b2 = 5 are given. What is loga(b – 1)?

A)	1	B)	1	C) 1	D) 2	E) 3
	3		2

8. Given x 1 and		xlogy x =		x, solve logx y – logy x
for x.
A) 3	B) 2	C)	5	D) 3	E)	7
2			2			2

9. Solve 5x + 1 = 2x for x.
A) log5 2		B) log2 5	C) log 5	2
			2
D) log 2	2	E) log2	5
5		5

10.What is the solution set for log(3x – 4) – log(3x + 1) = 0?

A)	{–	1}	B)	{4}			C)
		3		3
		D)	E) – {–		1	,	4}
					3		3

Chapter Review Test 3G

249

11.What is the sum of the solutions of

3log x + 32 – log x = 10?

A)	101	B) 11 C)	1001	D) 101 E) 1001
	100		10

12.9x – (10 3x + 1) + 34 = 0 has solutions x1 and x2. What is logñ2(x1 + x2)?

A) 8	B) 4	C) 2	D)	1	E)	1
				2		4

13.What is the sum of the integers in the solution set for log2(log4(x – 6)) < 0?

A) 36

B) 33

C) 24

D) 19

E) 17

14.How many natural numbers satisfy the inequality log3(3x – 5) log9(x + 6)2?

A) 3

B) 4

C) 5

D) 6

E) 7

15.What is logx + 1 (x + 9) if x satisfies log3(2x + 1) – log9(x – 2)2 = 1?

A)	4	B)	5	C) 2	D)	7	E) 3
	3		3			3

16.		log 8			2 log ñx
K	M			L	N
	log(2x + 1)		1 3
		3 log(	3	ò54)

In the figure above,

KL = log 8, LN = 2 log ñx,

KM = log(2x + 1) and MN = 3 log(13 3 54). What is x?

A)	1	B)	1	C) 1	D) 2	E) 5
	2		5

17. If log7(2x – 7) – log7(x – 2) = 0, what is log5 x?

A) 0

B) 1

C) 2

D) 3

E) 4

18.Which of the following is the solution set for log3(9 3x + 3) = 3x + 1?

A) {–1, 1}	B) {0, 2}	C) {0}
D) {1}	E) {2}

19. Solve 4log8 x3 = 2x+3.

A) {–3} B) {–1} C) {1} D) {2} E) {3}

20.What is the product of the solutions of the equation 4x + 164x =10?

A)	3	B)	5	C) 3	D)	15	E) 3
	4		4			4

250	Algebra 11

CHAPTER REVIEW TEST 3H

1. What is the area of triangle				y	f(x) = log4 x
1. What is the area of triangle				y	f(x) = log4 x
ABC in the figure?							A
					C		B
					C		B
			0		1 60	64		x
A) 32	B) 16	C) 6	D) 12				E) 24

2. If log3(log7 (log5 x)) = 0, what is x?
A) 0	B) 1	C) 73	D) 75	E) 57

3. Solve 71+log49 ( x+1) = 35.

A) {6} B) {12} C) {24} D) {48} E) {64}

4. Which value of x satisfies 5x = 3x + 1?
A) log3 5		B) log5 3	C) log3	3
			5
D) log5	3	E) log15 3
3

5. Solve log3(3 + 3 log3 x) = 3.

A) {32} B) {34} C) {35} D) {36} E) {38}

6.What is the solution set for the equation 3 log5 x + logx 5 = 4?

A) {3 5, 5}	B) {3 2, 2}	C) {ñ5, 5}
D) {ñ2, 2}	E) {3 3, 3}

7.How many natural numbers satisfy the inequality log5(x – 2) 2?

A) 23

B) 24

C) 25

D) 26

E) 27

8.Which value of x satisfies

2x – log(52x + 4x – 16) = x log 4?

A) 1

B) 2

C) 3

D) 4

E) 5

9. x1 and	x2 are	the	two	roots of	the	equation
(x2 +5)log3 ( x2 5) = 81. Which of the following is a
possible value of		x x2	?
		1
A) 16	B) 9	C) 1		D)	1	E)	1
A) 16	B) 9	C) 1		D)	9	E)	27
			4		9		27

						1
10. What is the sum of all the solutions of x							logx 5	= 5?
A)	11	B)	126	C)	26	D) 6		E) 11
	5		25		5

Chapter Review Test 3H

251

11. How many integers satisfy

log3(x+1) – log 1(3 – x) 1?

A) 3

B) 4

C) 5

D) 6

E) 7

12. Which value of a satisfies								5 ?
log	4	5 log		6 log	7 ... log	3a + 1	(3a + 2) =	5 ?
	4		5	6		3a + 1		2
								2
A) 8			B) 10		C) 11		D) 12	E) 15

13. Which value of x satisfies log 2				(x+3) 1?
			5
A) 1 B) –2 C)	–	14	D)		–	16	E)	–	46
		5				5			5

14. Solve log3	2	4 = log	3	(9 x ).
	2		3
A) {91}	B) {94}			C) {49} D) {4} E) {9}

15.What is the sum of the solutions of log6(x + 2) + log6(x + 7) = 2?

A) 2

B) 3

C) 4

D) 5

E) 6

16. What is the solution set for				ex – e– x				=	1	?
16. What is the solution set for				e	x	+ e	– x	=	2	?
				e		+ e			2
A) {	1 ln3, –	1 ln3}	B) {ln ñ3}					C) {2 ln 3}
	2	2
	D) {1 ln 3}		E) {–ln ñ3}
	2

17.What is the product of the solutions of log3(10 – 3x) = 10log(2 –x)?

A) –2

B) –1

C) 0

D) 1

E) 2

18.What is the solution set for the equation log3 x logx 5 = log3 5?

A) {3, 5}	B) {3, 5, 15}	C)
D) – {1}	E) + – {1}

19.Which value cannot be taken by x if

(x – 1)log2 (16– x2 ) =1?

A) –ò15

B) ò15

C) 0

D) 2

E) 4

20. What is the solution set for log2(x2 – 1) 3?

A) (1, 3] B) [–3, –1) C) [–3, 3]

D) (–1, 1) E) [–3, –1) (1, 3]

252	Algebra 11

CHAPTER REVIEW TEST 3K

1. Solve log2(log3 x) = 4 for x.

A) {38} B) {312} C) {316} D) {27} E) {212}

2. Solve 3log9 x = 3 for x.

A) {9} B) {3} C) {ñ3} D) {3 3} E) {4 3}

3. Which ordered pair satisfies the system

	5y+1 – 3x+1 =116	?
	5y +3x = 28	?
	5y +3x = 28
	A) (–2, 1)	B) (1, 2)
	D) (–1, 2)	E) (2, –1)

4. Solve 10ln x + (3 xln 10) = 40 for x.

2	1				1
A) {e	} B) {e} C) {		}	D)	{		}
		e
						e2

C) (2, 1)

E) {e4 }

5.What is the sum of the solutions to the equation 3 log2 x – log2 8x+1= 0?

A) 2

B) 8

C) 14

D) 16

E) 18

6.How many integers satisfy the inequality log2(4x – 2) 3?

A) 1

B) 2

C) 3

D) 4

E) 5

7.What is the product of the values of x which satisfy 2 log2 x + logx 2 = 3?

A)	1	B) ñ2	C) 2	D) 2ñ2	E) 4
	2

8. Solve 2ln a – (3 aln 2) = –8 for a.

	1			1	2	}
A)	{		}	B) {e}	C) {1} D) {e} E) {e
		e2

9. Find the sum of the solutions to the equation

log2 x = log2		x.
A) 3	B) 5	C) 9	D) 17	E) 33

10. Solve log (x – 4) + log		(x2	+ 8x + 16) = 2.
3	9
A) {5}	B) {6} C) {7} D) {9} E) {16}

Chapter Review Test 3K

253

11.Which of the following is the solution set for log3(3 – x) 2?

A) (–9, 6]	B) [–6, 3)	C) (–6, –3)
D) [–3, 6)		E) (–9, 3)

12.How many integer solutions does (log5 x)2 – 4 0 have?

A) 25

B) 26

C) 27

D) 28

E) 29

13.What is the solution set for

logx 2 + logx 4 + logx 8 + ... + logx 256 = 12?

A)	{	1}	B) {2} C) {4} D) {8} E) {12}
		2

14.Which value of x satisfies the equation ln(x + 1) – ln x = –1 + ln3?

A)	3		B)	3		C)	e
	e+3			e – 3			3 – e
	D)	e		E)	3e

		e+3			3 – e

15. What is the solution set for logx(2x – 4) 1?
A) (0, 2]			B) (2, 4]	C)	[	1	, 2)
	1					2
D) (	1	, 4]	E) (3, 5]
	2

16.(13 6x) – (6 4x) – (6 9x) = 0 has solutions x1 and x2. Find 5x1 x2 .

A) 25	B) 5	C) 1	D)	1	E)	1
				5		25

17. Solve log	3	(7 – log	2	( x))= 2.
	3		2	3
A) {1}	B) {1}			C) {2}	D) {3} E) {3}
4		3		3	4

18. Which value of a satisfies					( a a)log	a 9 =	27?
A)	4	B)	3	C)	5	D) 2	E)	5
	3		2		3			2

19. Which value of x satisfies log x – 5 log 3 = –2?

A) 1.25	B) 0.81	C) 2.43
D) 0.8	E) 0.8 or 1.25

20. Which values of x satisfy 2 log(2x – 100) < 3?

A) 50 x < 250	B) 100 x < 120
C) 250 x < 500	D) 300 x < 550

E) 100 x < 550

254	Algebra 11

CHAPTER REVIEW TEST 3L

1.x and y are integers such that

(x log400 5) + (y log400 2) = 3. Calculate x + y.

A) 13

B) 15

C) 18

D) 20

E) 21

2. Which value of x satisfies

2log5 x2	– 21+log 5 x +2–1+log 5 x =1?
A) 25	B) 5	C) 1	D)	1	E)	1
				5		25

3. In the triangle opposite,						A
[AD] [BC], [AB] [AC],
AB = log y,
AC = log 3,				B				C
AD = log x and BC = log 9.
AD = log x and BC = log 9.						D
What is	y	?
What is		?
	x
A) x2		B) x	C) 1	D)	1		E)	1
A) x2		B) x	C) 1	D)	x		E)	x2
					x			x2

4. What is the sum of the solutions to (ln x)2 = ln x2?

A)	1	+1	B) e – 1	C)	1	+1
A)	e2	+1	B) e – 1	C)	e	+1
	e2				e
		D) e2 + 1		E) e(e + 1)

5.Solve xlog6 3 +3log6 x =18.

A) {3} B) {6} C) {18} D) {36} E) {54}

6.How many integer values of x satisfy the inequality –1 < log2(log4(x – 2)) < 1?

A) 16

B) 15

C) 13

D) 11

E) 10

7. What is the sum of the values of x and y which

satisfy the system log3 x5 – log5 y3 = 7 ?log3 3 x+ log 5 y= 8

A) 15

B) 14

C) 13

D) 10

E) 9

8. What is the solution set for 1 + logx 3 > 2 logx 4?

A) (– ,	16)	B) (1,			16)	C) (16	, )
	3				3	3
D) (0, 1) (		16	,	)		E) (0, 1)
		3

9.x 1 and logxy yx + log y2 x=1 are given. What is logxy?

A) –	1	B) –2	C)	1	D) 2	E) 4
	2			2

10.How many different integers satisfy the inequality log(x2 + 21) < 1 + log x?

A) 3

B) 4

C) 5

D) 6

E) 7

Chapter Review Test 3L

255

11.What is the product of the solutions to log2(100x)+ log 2(10 x)=14+ log x1?

A) 10–1/2	B) 10–3/2	C) 10–5/2
D) 10–7/2	E) 10–9/2

12. Solve (3 log 2) + 2 – x = log (3x – 52 – x).
5	5
A) {6} B) {5}	C) {4} D) {3} E) {2}

13.Which of the following is the solution set for xln x e?

A) [	1	, e]	B) [	1	, e]			C) [1, 3]
A) [	e	, e]	B) [	e2	, e]			C) [1, 3]
	e			e2
		D) [1, e2]			E)	[	1	, 1]
							e

14. Given log x + log100 x = 7, calculate x3/2.

A) 1014/3 B) 107 C) 106 D) 105 E) 104

15.What is the product of the solutions to (ln x)2 – ln x5 – ln e6 = 0?

A) e6

B) e5

C) e–6

D) e–5

E) e–1

16. Solve (2x)logb 2 – (3x)log b 3 = 0 for x > 0 and b > 0.

A) {	1	}	B) {1} C) {1} D) {6} E) {12}
	216
			6

17. Which value of x satisfies 22x – (8 2x) + 12 = 0?

A) log 3	B)	1 log 6		C) 1+ log	3
		2			2
D) 1 + log2 3			E)	log 3
				log 2

18. What is the solution set for xlog x = x3 ? 100

A) {	1	}	B) {10}		C) {100}

10
	D) {10, 100}		E) {	1	, 10}

			10

19. Find the sum of the solutions to x3– log6 x = 36.

A) 42 B) 41 C) 40 D) 36 E) 70

20.What is the solution set for the inequality log5(x + 1) + log5(x – 1) log5 3?


A) –2 x 2 B) –2 < x < 2		C) –1 < x 2
D) 1 x 2	E) 1 < x 2

256	Algebra 11

CHAPTER 4

A. WRITING EQUATIONS IN QUADRATIC FORM

Definition

EXAMPLE 1

Solution 1

standard form of an equation

An equation is in standard form if the only term on the right-hand side of the equation is zero.

For example, the equations 6x2 + 2x – 3 = 0 and x4 – 5 = 0 are both in standard form. The equation 6x2 + 2x = 3 and x4 = 5 are not in standard form.

Certain equations that are not quadratic can be expressed in quadratic form using substitutions. These equations can be recognized because when they are written in standard form, the exponent of the variable in one term is half the exponent of variable in the other term.

For example, we can write standard form equations such as x4 + 17x2 + 72 = 0

2x8 + 4x4 = 0 x – ñx – 12 = 0

as quadratic equations, because the exponent of the first variable is twice the exponent of the second variable.

Look at the steps to write an equation as a quadratic.

1.Let t be a variable term with the half exponent.

2.Substitute t in all the terms with the variable.

3.Solve for t.

4.Back substitute for the original variable.

Solve x4 – 13x2 + 36 = 0.

The equation x4 – 13x2 + 36 = 0 is not a quadratic equation but we can write it as (x2)2 – 13x2 + 36 = 0. For this reason, it is a quadratic in x2. Let x2 = t.

First we solve for t, then solve the resulting equations for x. (x2)2 – 13x2 + 36 = 0

x2 = t, so t2 – 13t + 36 = 0. By factoring, (t – 4)(t – 9) = 0

t = 4 or t = 9. Since t = x2

x2 = 4		x2 = 9
x = 2	or	x = 3 .

258	Algebra 11

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