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Logarithmic Equation

Solve the following equation:

a) Log10 (x^{2} – 4x + 7) = 2

b) Log_{8} (r^{2} – 8r + 18) = 1/3

Solution

a) Log_{10} (x^{2} – 4x + 7) = 2

x^{2} – 4x + 7 = 10^{2} (index form)

x^{2} – 4x + 7 = 100

x^{2} – 4x + 7 – 100 = 0

x^{2} – 4x – 93 = 0

Using quadratic formula

x = __– b ±√b ^{2}– 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) Log_{8} (x^{2} – 8x + 18) =1/3

x^{2} – 8x + 18 = 8^{1/3}

x^{2} – 8x + 18 = (2)^{3X1/3}

x^{2} – 8x + 18 =2

x^{2} – 8x 18 – 2 = 0

x^{2} – 8x + 16 = 0

x^{2} – 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_{b}P = x and this means P = b^{x}

Log_{c}P = log_{c}b^{x} = x log_{c}b

If x log_{c}b = log_{c}P

x = __log _{c}P__

log_{c} b

:. log_{c}P = __log _{c}P__

log_{c}b

Example :

Shows that log_{a}b x log_{b}a = 1

Log_{a}b = __log _{c}b__

log_{c}a

Log_{b}a = __log _{c}a__

log_{c}b

:. log_{a}b x log_{b}a = log_{c}b x log_{c}a

log_{c}a + log_{c}b = 1

Evaluation

Solve (i) Log_{3} (x^{2} + 7x + 21) = 2 (ii) Log_{10} (x^{2} – 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 10^{n} 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 = 10^{4}

Number Power of 10

100 10^{2}

10 10^{1}

1 10^{0}

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 10^{n}, 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 10

^{3}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 10^{3}

2) 382 = 3.82 x 10^{2}

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 x10^{5}

628000 = 10^{0.7980} x 10^{5}

= 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 10^{3}

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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