LOGARITHMIC FUNCTIONS-1

If F(x) = log₅x, Find F(¹/₁₂₅)

Solution

F(x) = log₅x

F(¹/₁₂₅) = log₅(¹/₁₂₅)

F(¹/₁₂₅) = log₃125^-1

F(¹/₁₂₅) = log₅(5³)^-1

F(¹/₁₂₅) = log₅5^{(3 x -1)}

F(¹/₁₂₅) = log₅5^-3

F(¹/₁₂₅) = -3log₅5

F(¹/₁₂₅) = -3 x 1

F(¹/₁₂₅) = -3

Therefore F(¹/₁₂₅) = - 3

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

If F(x) = log₃x, Find F(¹/₇₂₉)

AXIS OF SYMMETRY 2

Find the axis of symmetry for F(x) = 6x² + 60x + 8

Solution

a=6, b=40

Axis of symmetry = -b/2a

                                   =  -(60)

                                       2 x 6

                                   =    -60

                                           12

                                    =   -5

                                            

Hence the axis of symmetry is -5 .

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

Find the axis of symmetry for F(x) = 5x² + 200x + 8

ALGEBRA - 4

Find w in the following equation;

2w - w-3 = w

5        2      7

solution

2w - w-3 = w

5         2      7

The LCM of 5, 7 and 2 is 70. So, we multiply by 70 throughout

70x 2w - 70(w-3) = w x 70

       5             2           5

¹⁴70x 2w – ³⁵70(w-3) = w x 70¹⁰

         ₁5            ₁2         7₁

14 x 2w - 35(w-3) = 10w

28w - 35w + 105 = 10w

-7w + 105 = 10w

105 = 10w + 7w

105 = 17w

105 = 17w

17      17

¹⁰⁵/₁₇ = w

Hence w = 105/17

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

Find a in the following equation:

3+a  -  a-8 = a

   5        2      5

FUNCTIONS 6

If F(x) = 4x² + 12x - 420; find F(10) - F(2).

solution

PROCEDURE:

i) Find F(10)

ii) Find F(2)

iii) Do F(10) - F(2).

i) Find F(10)

F(x) = 4x² + 12x - 420

F(10) = 4(10)² + 12(10) - 420

F(10) = 4(10 x 10) + (12 x 10) - 420

F(10) = 400 + 120 - 420

F(10) = 520 - 420

F(10) = 100.

ii) Find F(2)

F(x) = 4x² + 12x - 420

F(2) = 4(2)² + 12(2) - 420

F(5) = 4(2 x 2) + (12 x 2) - 420

F(5) = 32 + 24 - 420

F(5) = 56 - 420

F(5) = -364

iii) Do F(10) - F(5).

      F(10) - F(5) = 100 - (-364) = 100 + 364 = 464.

Hence F(10) - F(5) = 464.

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

If F(x) = 5x² + 7x - 10; find F(10) - F(4).

LOGARITHMS-8

Evaluate Log₂(256 ÷ 8).

Solution

= Log₂(256 ÷ 8)

= Log₂256 -  Log₂8          (applying the quotient rule)

= Log₂2⁸ -  Log₂2³             ( 256= 2⁸ and 8=2³ )

= 8Log₂2 -  3Log₂2       ( remember  Log_aa^c = cLog_aa )

= (8 x 1) - (3 x 1)          ( remember  Log_aa = 1 )

= 8 - 3

= 5

hence Log₂(128 ÷ 8) = 5 

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

Evaluate Log₂(1024 ÷ 32). 

SETS 1

If n(A)= 170 , n(B)= 160 and n(AnB)=150, find n(AuB).

Solution

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

             = 170 + 160 – 150

             = 330 – 150

             = 180

Hence n(AuB) = 180 answer

TRY THIS…………………………….

If n(A)= 146 , n(B)= 164 and n(AnB)=130, find n(AuB).

LOGARITHMIC EQUATIONS-4

Find x if Log_x256 = 4

solution

Log_x256 = 4

256 = x⁴ .

4⁴ = x⁴ . [since 256 = 4⁴ by prime factorization]

4⁴ = x⁴ . [ powers cancel out]

x = 4

Hence x = 4

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

Find y if Log_y64 = 3

VECTORS-2

If u = 6i + 2j and v = 3i + 10j find 2u - 3v

solution

= 2u - 3v

= 2(6i + 2j) - 3(3i + 10j)

= 12i + 4j - 9i - 30j

= (12i + 9i) + (4j - 30j)

= 21i - 26j

Hence 2u - 3v = 21i - 26j

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

If u = 9i + 3j and v = 2i + 4j;  find 9u - 2v.

FUNCTIONS-5

If F(x) = 4x² + 12x - 420; find F(6) - F(4).

solution

PROCEDURE:

i) Find F(6)

ii) Find F(4)

iii) Do F(6) - F(4).

i) Find F(6)

F(x) = 4x² + 12x - 420

F(6) = 4(6)² + 12(6) - 420

F(6) = 4(6x 6) + (12 x 6) - 420

F(6) = 144 + 72 - 420

F(6) = 216 - 420

F(6) = -204.

ii) Find F(2)

F(x) = 4x² + 12x - 420

F(4) = 4(4)² + 12(4) - 420

F(4) = 4(4 x 4) + (12 x 4) - 420

F(4) = 64 + 48 - 420

F(4) = 112 - 420

F(4) = -308

iii) Do F(4) - F(2).

      F(6) - F(4) = -204 - (-308) = -204 + 308 = 104.

Hence F(4) - F(2) = 104.

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

If F(x) = 15x² + 2x - 3; find F(8) - F(3).

LOGARITHMS-9

If log₅(3x + 101)=3; find x

   Solution

log₅(3x + 101)=3

 (3x + 101)=5³   

                      

 3x + 101=125

3x = 125 – 101

3x =  24

3x =  24   

3        3

X = 8

Hence x = 8

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

If log₂(2x - 14)=5; find x