LOGARITHMS H8

If Log_ay = Log_a26 +  Log_a6 ; find y.

 

Solution

 

Log_ay = Log_a26 +  Log_a6

 

Log_ay = Log_a(26x 6)         [since Log_am +  Log_an = Log_a(m x n)] 

 

Log_ay = Log_a156

 

Hence y = 156                 [since Log_a cancels out]

 

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

 

NECTA 1997 QN 13a

 

 

 

If Log_ax = Log_a5 +  Log_a3 ; find x

  

VARIATION H5

x is inversely proportional to y. x=18 while y=5. Find y when x is 15.

 

Solution

 

x ⍺ k

      y

 

x = k

      y

 

18 = k

        5

 

k = 18 x 5

 

k = 90

 

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

 

k = 90, y= ?, x = 15.

 

x = k

     y

 

15 = 90

        y

 

y x 15 = 90  x  y   (multiply by y on both sides)

              y

 

 

15y = 90

15      15

 

Hence y = 6  .

 

 

TRY THIS…………….

 

 

x is inversely proportional to y. x=80 while y=4. Find y when x is 10.   

LOGARITHMS H7

 

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

 

=(4x1) – 3(0.3010)

 

=4– 3(0.3010)

 

=4 – 0.9030

 

=3.097

 

Hence log1250=3.097

 

 

TRY THIS……………

 

 

If log 2= 0.3010; find the value of log 5,000 without using tables.

EXPONENTIALS G7

 

If y⁵ = 11, find y¹⁰ - 4y⁵.

 

Solution

 

= y¹⁰- 4y⁵.

 

= (y⁵)²- 4y⁵

 

= (11)² – (4 x 11)     remember y⁵ = 11

 

= (11 x 11) – (4 x 11)

 

= 121 – 44

 

= 77

 

Hence y¹⁰- 4y⁵= 77.

 

TRY THIS……………….

 

 If  w⁹ = 5, find w²⁷- 5w⁹.

ARITHMETIC PROGRESSION H6

 

The 1st term of an A.P. is 200 and the common difference is 80. Find the 10th term.

 

Solution

 

A₁= 200, d = 80

 

A_n = A₁ + (n-1)d

 

A₁₀ = A₁ + (10-1)d

 

A₁₀ = A₁ + 9d

 

A₁₀ = 200 + (9x80) [after substituting A₁= 72, d = 80 as given above]

 

∴ A₁₀ = 200 + 720

 

A₁₀ = 920

 

Hence the 10th term is 920.

 

  

TRY THIS……………

 

 

 

The 1st term of an A.P. is 600 and the common difference is 130. Find the 5th term.

VARIATION H3

 

x is directly proportional to y. x=32 while y=8. Find y when x is 184.

 

Solution

 

x ⍺ y

x = ky

 

32 = k x 8

 

32 = 8k    [dividing by 8 both sides]

8      8

 

k = 4

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

 

k = 4, y= ? x = 204.

 

x = ky

 

204 = 4 x y

 

204 = 4y

4        4

 

y = 51.

 

TRY THIS…………….

 

 

x is directly proportional to y. x=40 while y=4. Find y when x is 190.

ALGEBRA H4

If │2x – 7 │= 37; Find  x

 

Solution

 

±(2x – 11 )= 37

 

2x -11= 37    OR   –(2x-11) = 37

 

2x – 11 = 37    OR   -2x+11=37

 

2x = 37 + 11    OR   -2x= 37-11

 

2x = 48     OR   -2x  =   26

2      2               -2        -2

 

x = 24  OR  x = -13

 

 

Hence x = 14 OR x = -3

 

TRY THIS………………………. 

 

 

If │2x – 15 │= 63; Find x 

SIMPLIFY G8

 

Express as a single fraction m - 3 + m+5

                                              8         2

solution

 

= m - 3 + m+5

    8          2

 

= m - 3 + 4(m+5)                [Since the LCM of 8 and 2 is 8]

            8         

 

= m - 3 + 4m+20

            8         

 

= m + 4m + 20 - 3

            8         

 

= 5m +  17

        6                       

 

= 5m +  17       answer

        6         

 

 

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

 

Express as a single fraction 7e - 1 + e+1

                                              3          6

FACTORIZE G9

 

Factorize 121-25m²

 

Solution

 

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

 

121-25m² = 11²-5²m²       

 

            = 11² - (5m)²       

 

            = (11 - 5m)(11 + 5m)

 

Hence 121-25m² = (11 - 5m)(11 + 5m)  answer

 

 

TRY THIS…………….

 

 

 

Factorize 64 – 49y²