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 = 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 4³ = 64^a-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 x²  + 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 x²  + 12x to make the expression a perfect square? 

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 log_aaⁿ  = nlog_aa]

= (7x1) + (-5x1) + (5x1)             [since log_aa = 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 5^2w (40^w) = 100,000,000 ; Find w.

Solution

5^2w (40^w) = 100,000,000

(5²)^w (40^w) = 10⁸

(25)^w (40^w) = 10⁸

(25 x 40)^w = 10⁸

(1000)^w = 10⁸

(10³)^w = 10⁸

10^3w = 10⁸   (Bases are alike, so they cancel out)

3w = 8

w = 3/8

TRY THIS…………………………….

If 4^2c (4^c) = 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

x₁ = -3,  y₁ =-4,  x₂ = 18,  y₂ = -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)

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 Log_ax = 41, find Log_a(¹/x²)

Solution

= Log_a(¹/x²)

= Log_ax^-2

= -2 x Log_ax

= -2 x 41

= -82

TRY THIS………..

  

If Log_ab= 300, find Log_a(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)