LOGARITHMS 11

Change the following into Logarithmic form.

i)  5³ = 125

ii)  3³ = 27

iii)  11⁰ = 1

Solution

5³ = 125 ⇔ log₅ 125 = 3

3³ =27 ⇔ log₃ 27 = 3

11⁰ = 1 ⇔ log₁₁ 1 = 0

TRY THIS………………….

Change the following into Logarithmic form.

i)  11² = 121

ii)  3⁴ = 81

iii)  15⁰ = 1 

FACTORIZING-3

Factorize 1 - 225m²

Solution

We use difference of two squares a² – b² = (a - b)(a + b)

1 - 225m² = 1² - 15²m²       

                = 1² – (15m)²       

                = (1 - 15m)(1 + 15m)

Hence 1 - 225m² = (1 - 15m)(1 + 15m) answer

TRY THIS…………….

Factorize 1 – 121c²

EXPONENTIALS 7

If 8²⁰ = 32^a-2; find a.

Solution

8²⁰ = 32^a-2 

(2³)²⁰ = (2⁵)^a-2 

2⁶⁰ = 2^5(a-2) 

2⁶⁰ = 2^5a-10

2⁶⁰ = 2^5a-10      [Same bases both sides, cancel out]

60 = 5a – 10

60 + 10 = 5a

70 = 5a

70 = 5a

5      5

14 = a

Hence a=14

TRY THIS……………………….

If 4⁷ = 64^y-2; find y.

LOGARITHMS 10

If log₅(2x + 19)=2; find x.

   Solution

Log₅(2x + 19)=2

(2x + 19)=5²           

                                            

2x + 19=25

2x = 25 – 19

2x = 6

2x =   6   

2        2

x = 3

Hence x = 3

TRY THIS………………

If log₇(2t - 23)=2; find t

WORD PROBLEMS 1

Ana, Joy and Fred shared 800 sweets such that Ana got 10 sweets less than Joy while Fred got three as much as what Ana got. Find those given to Fred.

Solution

The starting point is Joy. Let the sweets given to Joy be x. See the table below:

ANA	JOY	FRED
x-10	x	3(x-10)=3x-30

  

Then;

ANA + JOY + FRED = 800

x-10 + x + 3x-30 = 800

x + x + 3x - 30 - 10 = 800

5x - 40 = 800

5x = 800 + 40

5x = 840

5x = 840

5       5

x = 168

The sweets given to Fred

= 3x – 30

= 3(168) – 30

= 504 – 30

=474

Hence Fred got 474 sweets.

TRY THIS………………………

Kemi, Roy and Isaya shared 610 sweets such that Kemi got 10 sweets less than Roy while Isaya got four as much as what Kemi got. Find those given to Kemi.   [100 answer] 

RATIOS 1

Given the ratios U:V = 9: 40 and V:W = 20:7 Find the ratio U:W.

solution

U = U x V

W   V   W

U =    9  x  20¹

W   ₂ 40    7

U =   9 x 1

W    2 x 7

U =   9

W    14

Hence U:W = 9:14

TRY THIS...............................

Given the ratios D:E = 20:7 and E:F = 28:15;  Find the ratio D:F.

PERCENTAGE PROFIT 2

Dikshi got a profit of 1,794/= after selling an item for 24,794/=. Find the percentage profit.

Solution

%’ge profit = Profit   x  100  [where B. P. represents Buying Price.]

                        B.P

But; Profit = SP - BP;     [where S. P. represents Selling Price.]

So; B.P = SP – Profit

B.P = 24,794 – 1794

       = 23,000

%’ge profit = 1794   x  100    

                      23000

%’ge profit = 1794 

                      230

%’ge profit = 7⁴/5%   

        

              

Hence Percentage Profit was 7⁴/5%   

TRY THIS………………..

Janhwi got a profit of 220,000/= after selling an item for 266,200/=. Find the percentage profit.      Answer 21%

AXIS OF SYMMETRY 1

Find the axis of symmetry for F(x) = 7x² - 50x + 7

Solution

a=7, b=-4

Axis of symmetry = -b/2a

                              =  -(-50)

                                  2 x 7

                              =    50

                                   14

                              =     25

                                     7

Hence the axis of symmetry is ²⁵/7

TRY THIS…………

Find the axis of symmetry for F(x) = 8x² - 15x + 7 

VECTORS 1

If u = 6i + 2j and v = 7i + 10j find 5u - 3v

solution

= 5u - 3v

= 5(6i + 2j) - 3(7i + 10j)

= 30i + 10j - 21i - 30j

= (30i + 21i) + (10j - 30j)

= 51i - 20j

Hence 5u - 3v = 51i - 20j

TRY THIS..................

If u = 7i + 3j and v = 4i + 4j;  find 2u + 6v.

LOGARITHMS 9

Evaluate Log₂(1024 x 8).

Solution

= Log₂(1024 x 8)

= Log₂1024 +  Log₂8          (applying the product rule)

= Log₂2¹⁰ +  Log₂2³             ( 1024= 2¹⁰ and 8=2³ )

= 10Log₂2 +  3Log₂2       ( remember  Log_aa^c = cLog_aa )

= (10 x 1) +  (3 x 1)          ( remember  Log_aa = 1 )

= 10 + 3

= 13

hence Log₂(1024 x 8) = 13 

TRY THIS...............

Evaluate Log₂(1024 x 64). 

EXPONENTIALS 6

If k⁵ - 200 = 2925; find k.

Solution

k⁵ - 200 = 2925

k⁵ = 2925 + 200 [taking 200 in the right hand side]

k⁵ = 3125

k = 5 [since the 5^th root of 3125 is 5]

Hence k = 3

TRY THIS…………………………

If w⁵ - 20 = 223; find w.

QUADRATICS 3

What must be added to x²  + 40x to make the expression a perfect square?

Solution

a=1, b=12,c=?

b² = 4ac

(40)² = 4 x 1 x c

1600 = 4c

1600 = 4c

 4         4

400 = c

Hence number to be added is 400

TRY THIS……………………

What must be added to x²  + 28x to make the expression a perfect square?  

LOGARITHMS 8

Simplify Log₂128 – Log₅3125 – Log₆216

Solution

= Log₂128 – Log₅3125 – Log₆216

= Log₂2⁷ – Log₅5⁵ – Log₆6³

= 7Log₂2 – 5Log₅5 – 3Log₆6  because Log_aaⁿ =  nLog_aa

= (7 x 1) – (5 x 1) – (3 x 1)    

= 7 – 5 - 3

= -1

Hence Log₂128 – Log₅3125 – Log₆216 = -1 

TRY THIS…………..

Simplify Log₂2048 – Log₃9 – Log₅3125

PROBABILITY 2

A number is chosen at random from 1 – 10 inclusive. Find the probability that it is a multiple of five or an even number.

Solution

THIS IS A MUTUALLY EXCLUSIVE EVENT.

Let n(S) represent sample space

P(E) = probability of even number

P(M) = probability of a multiple of 5

n(S) = {1, 2, 3, 4, 5, 6, 7, 8, 9,10} = 10

n(E) = {2, 4, 6, 8, 10} = 5

n(M) = {5,10} = 2

P(E) = ⁵/10

P(M) = ²/10

………………………….

P(EuM) = P(E) + P(M)

P(EuM) = ⁵/10 + ²/10

∴ P(EuM) = ⁷/10

TRY THIS………………

A number is chosen at random from 17 – 24 inclusive. Find the probability that it is a prime or an even number.

VARIATION 2

x is directly proportional to y. x=12 while y=4. Find y when x is 600.

Solution

x ⍺ y

x = ky

12 = k x 4

12 = 4k

4      4

k = 3

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

k = 3, y= ? x = 600.

600 = 3 x y

600  = 3y       (dividing by 3 on both sides)

 3        3

Hence y = 200 answer.

TRY THIS…………….

x is directly proportional to y. x=20 while y=4. Find y when x is 70.

FUNCTIONS 4

If F(x) = 21x + 10; Find F^-1(x).

Solution

HINT: F^-1(x) means inverse.

PROCEDURE:

Make x the subject and then interchange x and y variables.

Let y=F(x)

So,  y= 21x + 10

y – 10 = 21x

y – 10  = 21x

   21        21

y – 10   = x

   21

x  =   y – 10     after rearranging

           21

 y^-1 = x – 10     after interchanging x and y variables.

             21

F^-1(x) = x – 10     after interchanging x and y variables.

                21

Hence, F^-1(x) = x – 10     

                             21

TRY THIS……………………………

If F(x) = 17x + 26; Find F^-1(x).