LOGARITHM – SOLVING PROBLEMS BASED ON LAWS OF LOGARITHM (CONT)

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Logarithmic Equation

Solve the following equation:

a)         Log10 (x² – 4x + 7) = 2

b)        Log₈ (r² – 8r + 18) = 1/3

Solution

a)         Log₁₀ (x² – 4x + 7) = 2

x² – 4x + 7 = 10² (index form)

x² – 4x + 7 = 100

x² – 4x + 7 – 100 = 0

x² – 4x – 93 = 0

Using quadratic formula

x          = – b ±√b²– 4ac

2a

a = 1, b = – 4, c = – 93

x = – (- 4) ± √(- 4) ² – 4 x 1 x (- 93)

2 x 1

= + 4 ± √16 + 372

2

= + 4 ± √388/2

= x = 4 +√ 388/2 or 4 – √388/2

x =  11.84 or x = – 7.85

2)         Log₈ (x² – 8x + 18) =1/3

x² – 8x + 18 = 8^1/3

x² – 8x + 18 = (2)^3X1/3

x² – 8x + 18 =2

x² – 8x 18 – 2 = 0

x² – 8x + 16 = 0

x² – 4x – 4x + 16 = 0

x(x – 4) -4 (x – 4) = 0

(x – 4) (x – 4) = 0

(x – 4) twice

x = + 4 twice

Change of Base

Let log_bP = x and this means P = b^x

Log_cP = log_cb^x = x log_cb

If x log_cb = log_cP

x = log_cP

log_c b

:.          log_cP = log_cP

log_cb

Example :

Shows that log_ab   x   log_ba  = 1

Log_ab = log_cb

log_ca

Log_ba = log_ca

log_cb

:.          log_ab   x log_ba  =  log_cb  x log_ca

log_ca  +  log_cb = 1

Evaluation

Solve (i) Log₃ (x² + 7x + 21) = 2 (ii) Log₁₀ (x² – 3x + 12) = 1

(iii) 5^2x+1 – 26(5^x) + 5 = 0 find the value of x (iii) 5^2x+1 – 26(5^x) + 5 = 0 find the value of x

Numbers such as 1000 can be converted to its power of ten in the form 10ⁿ where n can be term as the number of times the decimal point is shifted to the front of the first significant figure i.e. 10000 = 10⁴

Number                                                 Power of 10

100                                                                              10²

10                                                                                 10¹

1                                                                                   10⁰

0.01                                                                             10^-3

0.10                                                                             10^-1

Note: One tenth; one hundredth, etc are expressed as negative powers of 10 because the decimal point is shifted to the right while that of whole numbers are shifted to the left to be after the first significant figure.

A number in the form A x 10ⁿ, where A is a number between 1 and 10 i.e. 1 < A < 10 and n is an integer is said to be in standard form  e.g. 3.835 x 10³ and 8.2 x 10^-5 are numbers in standard form.

Examples

Express the following in standard form

1)  7853   2)  382      3) 0.387   4)  0.00104

Solution

1)  7853 = 7.853 x 10³

2)  382 = 3.82 x 10²

3)  0.387 = 3.87 x 10^-1

4)  0.00104 = 1.04 x 10^-3

 

Base ten logarithm of a number is the power to which 10 is raised to give that number e.g.

628000 = 6.28 x10⁵

628000 = 10^0.7980 x 10⁵

= 10^{0.7980+ 5}

= 10^5.7980

If a number is in its standard form, its power is its integer i.e. the integer of its logarithm e.g. log 7853 has integer 3 because 7853 = 7.853 x 10³

Examples: Use tables (log) to find the complete logarithm of the following numbers.

(a)  80030    (b) 8      (c) 135.80

Solution:

(a)  80030  =          4.9033

(b)  8          =          0.9031

(c)  13580  =          2.1329

Multiplication and Division of number greater than one using logarithm

To multiply and divide numbers using logarithms, first express the number as logarithm and then apply the addition and subtraction laws of indices to the logarithms. Add the logarithm when multiplying and subtract when dividing.

Examples: Evaluate using logarithm.

1.   4627 x 29.3

2.   8198 ÷ 3.905

3.   48.63  x  8.53

15.39

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