LOGARITHMS 22

If log_x 2401 – log₃729 = -4; find x

solution

log_x 2401 – log₃729 = -2

log_x 2401 – log₃3⁶ = -2

log_x 2401 – 6log₃3 = -2

log_x 2401 – 6 x 1 = -2

log_x 2401 – 6 = -2

log_x 2401 = -2 + 6

log_x 2401 = 4 

2401 = x⁴        2401=7x7x7x7=7⁴ by prime factors

7⁴ = x⁴      Powers cancel out

x = 7

Hence x = 7.

TRY THIS………….

If log_m2048 – log₆216 = 8; find m

EXPONENTIALS 10

Simplify 44

               8^-2/3

Solution

=  44

    8^-2/3

= 44 x 1

           8^-2/3

=  44 x 8^2/3        [Since 1/a^-n = aⁿ]

=  44 x (8^1/3)²        [Since 8^1/3 = cube root of 8 = 2]

=  44 x (2)²        

=  44 x 4

= 176       

          

Hence   44        = 176

             8^-2/3

TRY THIS…..  

simplify 20

             125^-2/3

ARITH. PROGRESSION 3

The 1st term of an A.P. is 80 and the common difference is 27. Find the nth term.

Solution

A₁= 80, d = 27

A_n = A₁ + (n-1)d

A_n = 80 + (n-1)27

A_n = 80 + 27n – 27

A_n = 27n - 53

Hence the nth term is 27n - 53.

TRY THIS……………

The 1st term of an A.P. is 24 and the common difference is 34. Find the nth term.

LOGARITHMS 21

If log 2= 0.3010; find the value of log 1250 without using tables.

Solution

=log1250

=log(10,000 ÷8)

=log10,000 –log8

= log10⁴ - log2³

= 4log10 - 3log2

=(4 x 1) - (3 x 0.3010)

= 4  -  0.9030

=3.097

Hence log 1250 =3.097

TRY THIS……………

If log 2= 0.3010; find the value of log 2500 without using tables.

ALGEBRA 12

If 7x + 5y - 56 = 0; find x-intercept.

Solution

x-intercept is when y=0.

7x + 5y -56 = 0

7x + 5(0) - 56 = 0

7x - 56 = 0

7x = 0+ 56

7x = 56

7      7

x= 8

Hence x-intercept= 8

TRY THIS………..

If 8x - 7y - 40 = 0; find x-intercept.

EXPONENTIALS 9

If 2^(6m-7) x 3⁽ⁿ⁺⁸⁾ = (3⁴⁴) x (2⁶⁵); Find m and n.

solution

equating equal bases;

2^(6m-7) =  2⁶⁵

2^(6m-7) =  2⁶⁵

6m - 7 = 65

6m  = 65 + 7

6m  = 72

¹6m  =  72⁵

₁6         6₁

m =  12

again for n we equate equal bases.

3⁽ⁿ⁺⁸⁾ = 3⁴⁴

3⁽ⁿ⁺⁸⁾ = 3⁴⁴

n + 8 = 44

n = 44 - 8

n = 36

Hence m = 12 and n = 36

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

If 8^(2m-17) x 11^(5n+40) = (11⁷⁰) x (8⁴³); Find m and n.

GEOM. PROGRESSION 3

Find the sum of the 1st nine terms of the geometrical progression 3+6+12+24+…….

Solution

G₁ = 3, r=2, n=9

S_n = G₁(rⁿ-1)/r-1

               

S₉ = 3(2⁹-1)/2-1

             

S₉ = 3(2⁹-1)/1

              

S₉ = 3(2⁹-1)

S₉ = 3(512-1)

S₉ = 3(511)

S₉ = 1533

Hence the sum of the 1st nine terms is 1533.

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

Find the sum of the 1st seven terms of the geometrical progression 5+10+20+40+…….

ALGEBRA 11

48 + 24 =56

        a

Solution

48 + 24 =56

        a

   24  = 56 - 48

   a

   24  =  8

    a

¹ a  x  24  =  8 x a

        ₁ a

24 = 8a

24  = 8a

8       8

3 = a

Hence a=3.

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

If 30 + 34 = 47; find m

           m

SETS 4

If n(A)= 85 , n(B)= 96 and n(AuB)= 130, find n(AnB).

Solution

n(AuB) = n(A) + n(B) - n(AnB)

130 = 85 + 96 - n(AnB)

130 = 181 - n(AnB)

130 - 181 = - n(AnB)

-51 = -n(AnB)

n(AnB) = 51 [after dividing by -1 both sides]

Hence n(AnB) = 51 answer

TRY THIS………….

If n(A)= 92 , n(B)= 95 and n(AuB)= 137, find n(AnB).

GEOM. PROGRESSION 2

The sum of the 1st five terms of a geometrical progression is 484. If the common ratio is 3 find the 4th term.

Solution

S₅ = 484, G₁ = ?, r=3, n=5, A₄=?

We use the summation formula to find the 1^st term.

S_n = G₁(rⁿ-1)/r-1

               

S₅ = G₁(r⁵-1)/r-1

               

484= G₁(3⁵-1)/3-1

               

484= G₁(243-1)

                  2

2 x 484= G₁(242)

                  

968= 242 G₁

⁴ 968   =  G₁

  242

G₁ = 4

Now we solve for the 4^th term.

G_n = G₁r^n-1

G₄ = G₁r^4-1

G₄ = G₁r³

G₄ = 4 x (3 x 3 x 3)

     = 108

Hence the 4th term is 108.

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

The sum of the 1st five terms of a geometrical progression is 484. If the common ratio is 3 find the 7th term.

ALGEBRA 10

Expand 8w(5w + 3 - w)

Solution

= 8w(5w + 3 - w)

= (8w x 5w) + (8w x 3) - (8w x w)

= 40w² + 24w - 8w²

= [40w² - 8w²] + 24w collecting like terms

= 32w² + 24w answer.

TRY THIS………..

Expand 4a(7a+ 12 - a)

GEOMETRY 1

If x-15⁰ and 4x + 55⁰ are complementary angles, find the value of x.

solution

Complementary angles add up to 90⁰.

so,  x-15⁰ + 4x + 55⁰ = 90⁰.

so,  x + 4x + 55⁰ - 15⁰ = 90⁰.

so,  5x + 40⁰ = 90⁰.

so,  5x = 90⁰ -  40⁰

so,  5x = 50⁰

so,  5x = 50¹⁰

     5       5

Hence x = 10⁰

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

If 3x - 23⁰ and 2x + 98⁰ are complementary angles, find the value of x.

EXPONENTIALS 8

If 5^2w (40000^w) = 100 ; Find w.

Solution

5^2w (40000^w) = 100

(5²)^w (40000^w) = 100

(25)^w (40000^w) = 100

(25 x 40000)^w = 100

(1000000)^w = 100

(10⁶)^w = 10²

10^6w = 10²   (Bases are alike, so they cancel out)

6w = 2

6w = 2 ¹

6      6 ₃

w = ¹/3 answer

TRY THIS…………………………….

NECTA 2004 QN. 4b

If 3^2t (4^t) = 6 ; Find t.

LOGARITHMS 20

If log 2= 0.3010; find the value of log 16,000,000 without using tables.

solution

=log16,000,000

=log(16 x 1,000,000)

=log16 + log1,000,000

=log2⁴ + log10⁶

=4log2 + 6log10

=4(0.3010) + (6 x 1)

=1.2040+  6

=7.2040

Hence log 16,000,000  = 7.2040 answer

TRY THIS……………

If log 2= 0.3010; find the value of log 320,000,000 without using tables.   

MAKING THE SUBJECT 1

If D = y(c + a)    make a the subject of the formula.

            (c – a)

Solution

  

D = y(c+ a)   

        (c – a)

(c – a) x D = y(c + a) x (w – a)¹   multiply by(w-a) on both sides.

                       (c – a)₁

(c – a) x D = y(c + a)

Dc – Da = yc + ya     [open the brackets on both sides].

Dc – yc = Da + ya   [collecting the terms containing a together]

Dc – yc = a(D + y)      [factoring a out on RHS]

Dc – yc = a(D + y) ¹               [Divide by (D + y) on both sides].

(D + y)         (D + y) ₁

Dc – yc = a

(D + y)        

Hence  a =   Dc – yc

                       D + y       

TRY THIS…………

If B = w(h + m)    make m the subject of the formula.

            (h – m)

VARIATIONS 3

x is inversely proportional to y. x=12 while y=5. Find y when x is 15.

Solution

x ⍺ k

      y

x = k

      y

12 = k

        5

k = 12 x 5

k = 60

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

k = 60, y= ?, x = 15.

x = k

      y

15 = 60

         y

y x 15 = 60  x  y   (multiply by y on both sides)

                y

  

15y = 60          [Dividing by 15 both sides]  

15      15

y = 4 .

TRY THIS…………….

x is inversely proportional to y. x=20 while y=4. Find y when x is 16.

EXPONENTIALS 7

If 8¹⁰ = 32^a-2; find a

Solution

8¹⁰ = 32^a-2 

(2³)¹⁰ = (2⁵)^a-2 

2³⁰ = 2^5(a-2) 

2³⁰ = 2^5a-10

2³⁰ = 2^5a-10      [Same bases both sides, cancel out]

30 = 5a – 10

30 + 10 = 5a

40 = 5a

40 = 5a

5      5

8 = a

Hence a=8

TRY THIS……………………….

If 4⁷ = 64^y-2; find y