Friday 30 June 2017

LOGARITHMS 5

Change the following into Logarithmic form.

i)   4-3 = 1/43=1/64

ii)   10-4 = 0.0001

iii)  25 = 32

Solution

i)  4-3 = 1/43=1/64 log4 (1/64) = -3

ii)  10-4 = 0.0001
log10 0.0001 = -4

iii)  25 = 32
log2 32 = 5

TRY THIS………………….

Change the following into Logarithmic form.
i)   4-3 = 1/43
ii)   10-6 = 0.000001
iii)  27 = 128

EXPONENTIALS 1


If 410 = 32a-2; find a

Solution

410 = 32a-2 

(22)10 = (25)a-2 

220 = 25(a-2) 

220 = 25a-10

220 = 25a-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 43 = 64a-2; find a

ALGEBRA 1


If 2w2 = 242; find w

Solution

2w2 = 242

2w2 = 242               [Dividing by 2 both sides]
2          2

w2 = 121

w = 11          Because the square root of 121 is 11.   

Hence w = 11         

TRY THIS……………………….


If 3y2 = 432; find y.

POLYGONS 1

A regular polygon has 17 sides. Find the total interior angles of that polygon.

Solution

n = 17

Total angles = (n - 2)1800

                   = (17 – 2)1800

                   = 17 x 1800

                   = 30600

Total angles = 30600

TRY THIS……………………….


A regular polygon has 58 sides. Find the total interior angles of that polygon.

QUADRATICS 1

What must be added to x+ 8x to make the expression a perfect square?

Solution

a=1, b=8,c=?

b2 = 4ac

(8)2 = 4 x 1 x c

64 = 4c
64 = 4c
4      4

16 = c
Hence number to be added is 16

TRY THIS……………………

What must be added to x+ 12x to make the expression a perfect square? 

Thursday 29 June 2017

UNITS OF DISTANCE 1

Convert 27734m into km.

Solution

1km  = 1000m
  ?     = 27734m

After cross-multiplying;

= 1 x 27734
      1000

=   27734
     1000

= 27.734 km

Hence 27734m = 27.734km

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

Convert 72949m into km.


LOGARITHMS 4

Evaluate Log10,000,000 +  log 0.00001 + log3243

Solution

= Log10,000,000  +  log 0.001 + log3243

= Log107 +  log 10-5 + log335

= 7Log10 +  (-5log 10) + (5log33)    [since logaa= nlogaa]

= (7x1) + (-5x1) + (5x1)             [since logaa = 1]

= 7 + (-5) + (5)

= 7

Hence Log10,000,000 +  log 0.00001 + log3243 = 7

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



Evaluate Log100,000,000 +  log 0.01 + log32187

EXPONENTIALS 1


If 52w (40w) = 100,000,000 ; Find w.

Solution

52w (40w) = 100,000,000

(52)w (40w) = 108

(25)w (40w) = 108

(25 x 40)w = 108

(1000)w = 108

(103)w = 108

103w = 108   (Bases are alike, so they cancel out)

3w = 8

w = 3/8

TRY THIS…………………………….


If 42c (4c) = 512 ; Find c.

LOGARITHMS 3


If log 2= 0.3010; find the value of log 500,000 without using tables.

solution

=log500,000

=log(1,000,000÷ 2)

=log1,000,000 – log2

=log106 – log2

=6log10 – log2

=(6x1) – (0.3010)

=6 – 0.3010

=5.699

Hence log50,000 = 5.699


TRY THIS……………


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

SLOPE OR GRADIENT 1

Find the slope of a line which passes through (-3, -4) and (18,-11)

Solution

x= -3,  y=-4,  x= 18,  y= -11

m = y2 –y1
        x2 – x1

m =   -11 –(-4)
          18 –(-3)

m =   -11 + 4
          18 + 3

m =    - 7
          21


Hence the slope is -7/21

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


Find the slope of a line which passes through (-1, -4) and (4,-15)

Monday 26 June 2017

LOGARITHMS 2

 If log 3= 0.4771; find the value of log 90,000,000 without using tables.

solution

log90,000,000=log(10,000,000 x 9)

=log10,000,000 + log9

=log107 – log32

=7log10 – 2log3

=(7x1) –  2(0.4771)

=7– 0.9542

= 6.0458

Hence log 90,000,000 = 6.0458


TRY THIS……………


If log 3= 0.4771; find the value of log 270,000 without using tables.

FACTORIZE 1


Factorize 81-V2

Solution

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

81-V2 = 92-V2       

        = (9 - V)(9 + V)

Hence 81 - V2 = (9 - V)(9 + V)


TRY THIS…………….



Factorize 64 - Z2

LOGARITHMS 1


If Logax = 41, find Loga(1/x2)

Solution

= Loga(1/x2)

= Logax-2

= -2 x Logax

= -2 x 41

= -82


TRY THIS………..
  

If Logab= 300, find Loga(1/b2)

FUNCTIONS 1

If F(x) = 8x + 10; Find F-1(42).


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= 8x + 10

y – 10 = 8x

y – 10  = 8x
   8          8

y – 10   = x
   8

x  =   y – 10     after rearranging
           8

 y-1 = x – 10     after interchanging x and y variables.
             8

F-1(x) = x – 10     after interchanging x and y variables.
                8

Now we calculate F-1(42) as hereunder;

F-1(90) = 42 – 10     
                  8

F-1(90) =  32      
                8

F-1(90) =  4      [after dividing 32 by 8]
               
Hence, F-1(32) =  4   .      
                             


TRY THIS……………………………


If F(x) = 7x + 20; Find F-1(14).

MID-POINT 1


Find the midpoint of a line from (7, 14) to (17, 10)

Solution

x1=7, x2=17, y1=14, y2=10.

Mid point = (x1 + x2, y1 + y2)
                         2             2

Mid point = (7 + 17, 14 + 10)
                        2              2

Mid point = (24, 24)
                     2    2

Hence midpoint = (12, 12)

TRY THIS……………………..



Find the midpoint of a line from (16, 2) to (14, 16)