LOGARITHMS 14

Evaluate Log₂(1024 ÷ 8).

Solution

= Log₂(1024 ÷ 8)

= Log₂1024 -  Log₂8          (applying the quotient 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

= 7

hence Log₂(1024 ÷ 8) = 7 

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

Evaluate Log₂(1024 ÷ 64). 

VARIATIONS 1

x is inversely proportional to y. x=14 while y=5. Find y when x is 7.

Solution

x ⍺ k

      y

x = k

      y

14 = k

        5

k = 14 x 5

k = 70

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

k = 70, y= ?, x = 7.

x = k

      y

7 = 70

       y

y x 7 = 70  x  y   (multiply by y on both sides)

             y

7y = 70

7y = 70        (dividing by 7 on both sides)

7      7

y = 10

TRY THIS…………….

x is inversely proportional to y. x=80 while y=4. Find y when x is 16.

A. P. 2

The 1st term of an A.P. is 70 and the common difference is 97. Find the nth term.

Solution

A₁= 70, d = 97

A_n = A₁ + (n-1)d

A_n = 70 + (n-1)97

A_n = 70 + 97n – 97

A_n =  97n + 70 – 97

A_n = 97n – 27       [since 70 - 97= -27]

Hence the nth term is 97n - 27.

TRY THIS……………

The 1st term of an A.P. is 108 and the common difference is 46. Find the nth term.

EXPONENTIALS 2

If 50^2w (40^w) = 1000000⁸ ; Find w.

Solution

50^2w (40^w) = 1000000⁸

(50²)^w (40^w) = (10⁶ ) ⁸

(2500)^w (40^w) = 10^6x8

(2500 x 40)^w = 10⁴⁸

(100000)^w = 10⁴⁸

(10⁵)^w = 10⁴⁸

10^5w = 10⁴⁸   (Bases are alike, so they cancel out)

5w = 48

5w = 48^14.4

5        5

w = 9.6 answer

TRY THIS…………………………….

If 4^3a (16^a) = 512 ; Find a.

SIMPLIFY 2

Simplify: 5x + 90y + 2y - 80x

solution

= 5x + 90y + 2y – 80x

= 5x– 80x + 90y + 2y

= -75x + 92y or 92y-75x

= 92y-75x answer

TRY THIS……………………… 

Simplify: 20x + 33y + 25y – 93x.

LOGARITHMS 13

Simplify Log₂ 8192 – Log₅3125

Solution

= Log₂ 8192 – Log₅3125

= Log₂2¹³ – Log₅5⁵ 

= 13Log₂2 – 5Log₅5   because Log_aaⁿ =  nLog_aa

= (13 x 1) – (5 x 1)      because Log_aa =1

= 13 – 5

= 8

Hence Log₂ 8192 – Log₅3125 = 8 

TRY THIS…………..

NECTA 1995 QN 24b

Simplify Log₂32 – Log₃9.

SEQUENCES 1

The nth term of a certain sequence is 11n-7. Find the 11^th term

Solution

A_n = 11n-7

But n=11

A₁₁ = 11(11)-7   [since n = 11]

A₁₁ = 121 - 7

A₁₁ = 114

TRY THIS……………………… 

The nth term of a certain sequence is 2n² – 11n. Find the 5^th term

SIMPLIFY 1

Simplify the expression w² - t²/ (w + t)

Solution

from difference of two squares p² - q² = (p - q)(p + q)

= w² - t²/ (w + t)

= (w + t)(w - t)

      w + t

= (w + t)(w - t)

      w + t

=  w - t    answer

      

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

Simplify the expression 

u² - y²/ (u + y)

ANGLES 1

If A and B are complementary angles such that A = 27° and B = x + 55° , find the value of x .

Solution

A + B = 90⁰. [Since complementary angles add up to 90⁰]

27⁰ + x + 55⁰ = 90⁰.

x + 82⁰ = 90⁰.

x = 90⁰ - 82⁰

x = 8⁰  answer.

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

If A and B are complementary angles such that A = 33° and B = x + 16° , find the value of x .

EXPONENTIALS 1

If 4¹⁰ = 32^a-2; find a.

Solution

4¹⁰ = 32^a-2 

(2²)¹⁰ = (2⁵)^a-2 

2²⁰ = 2^5(a-2) 

2²⁰ = 2^5a-10

2²⁰ = 2^5a-10  [Same bases both sides cancels out]

20 = 5a – 10

20 + 10 = 5a

30 = 5a

30 = 5a

5      5

6 = a

Hence a=6

TRY THIS……………………….

If 5²⁰ = 625^W-2; find W