Logarithmic Properties

3.      Logarithmic Properties

Previously, we've talked about understanding the logarithm of a number, that is if  x = aⁿ , for a > 0 and a ≠ 1 then n = ^alog x. By considering the section G.1 , of course you can understand that

a ^{alog x} = x
^alog aⁿ = n

In addition to the above two properties, the logarithm also apply the following properties.

For a > 0, a ≠ 1, x > 0, y > 0, b > 0, and c > 0 , applies

a.       ^alog a = 1

b.      ^alog (x × y) = ^alog x + ^alog y

c.       ^alog  = ^alog x - ^alog y

d.      ^alog xⁿ = n × ^alog x

e.       1.  ^alog x = ^blog x/ ^blog a , b ≠ 1

2.  ^alog x = 1/ ^xlog a

f.       ^alog x × ^xlog y = ^alog y

g.       ^alog x = ^alog y if and only if x = y

h.      1.  ^{a^n}log x^m = ^alog x^m/n =  × ^alog x

2.  ^{a^n}log xⁿ = ^alog x

i.        ^alog  = - ^alog

Proof of these properties are as follows:
a.  ^alog a = 1
    Proof: suppose ^alog a = x . Based on the definition of logarithms, is obtained

    a^x = a  x = 1

    So, ^alog a = 1                                            (valid)

b.  ^alog (x × y) = ^alog x +  ^alog y

    Proof: a ^{alog x × y} = x × y

                         = a ^{alog x} × a ^{alog y}

                                  = a ^{alog x + alog y}

    Because a ^{alog x × y} = a ^{alog x + alog y} then ^alog x × y = ^alog x + ^alog y                              (valid)

c.  ^alog  = ^alog x - ^alog y

    Proof: a ^{alog x/y} =

                        = a ^{alog x}/a ^{alog y} = a ^{alog x - alog y}

     Because a ^{alog x/y} = a ^{alog x - alog y} so ^alog  = ^alog x - ^alog y                                           (valid)

d.  ^alog xⁿ = n × ^alog x

    Proof: a ^{alog x^n} = xⁿ

                         = (a ^{alog x})ⁿ

 = a ^{n × alog x}

                      Because a ^{alog x^n} = a ^{n × alog x} then ^alog xⁿ = n × ^alog x                                                     (valid)

            e. ^alog x = ^blog x/^blog a and ^alog x = 1/ ^xlog a

                1. ^alog x = ^blog x/^blog a

                    Proof:

                    Let ^blog x = m and ^blog a = n. Mean, x = b^m or b = x^1/m and a = bⁿ or b = a^1/n

                    Therefore, a ^{blog x/blog a} = a^m/n

                                                     = (a^1/n)ⁿ

                                                     = b^m

                                                     = x

                                                     = a ^{alog x}

                        Because a ^{blog x/blog a} = a ^{alog x} then ^alog x = ^blog x/^blog a                                         (valid)

                 2. ^alog x = 1/ ^xlog a

                     Proof:

                     From the previous properties, has been demonstrated that

                     ^alog x = ^blog x/^blog a .

                     If it is taken b = x then ^alog x = ^xlog x/^xlog a

                                                                   = 1/ ^xlog a                                                  (valid)

           

f. ^alog x × ^xlog y = ^alog y

               Proof: ^alog x × ^xlog y             = log x/log a × log y/log x

                                                = log y/log a = ^alog y                                                    (valid)

            g. ^alog x = ^alog y if and only if x = y

               Proof: ^alog x = ^alog y  a ^{alog x} = a ^{alog y}

                                                  x = y                                                          (valid)

            h. ^{a^n}log x^m = ^alog x^m/n =  × ^alog x and  ^{a^n}log xⁿ = ^alog x

               1. ^{a^n}log x^m = ^alog x^m/n =  × ^alog x

                  Proof: let ^{a^n}log x^m = p. Consequently, (aⁿ)^p = x^m

                                                                         a^{n × p} = x^m

                                                                         a^p = x ^m/n

                                                                         ^alog x^m/n = p

                                                                         ^alog x^m/n = ^{a^n}log x^m                                                 (valid)

                 By using properties d, ^alog xⁿ = n × ^alog x then ^alog x^m/n =  × ^alog x

                 So, proved that  ^{a^n}log x^m = ^alog x^m/n =  × ^alog x

              2. ^{a^n}log xⁿ = ^alog x

                 Proof: . ^{a^n}log xⁿ =  × ^alog x

                                         = ^alog x                                                        (valid)

i.        ^alog  = - ^alog

Proof:

                  ^alog = ^alog ()^-1

                           = -1 × ^alog

                           = -^alog

                 So, ^alog  = -^alog                                                          (valid)

Examples of application of the properties  of logarithms above are as follows.

Example :

1.      Specify the value of the logarithm ⁶log 9 + ⁶log 4.

Solution :

⁶log 9 + ⁶log 4    =  ⁶log ( 9 × 4 )

                                     =     ⁶log 36

                                     =     ⁶log 6²

                                          =      2

2.      If  ²log 5 = p, declare logarithms below in p.

a.       ²log 125                       b.    ¹⁶log 25

Solution :

a.   ²log 125                  =      ²log 5³

                                                   =   3 ²log 5

                                                 =   3p

b.      ¹⁶log 25                   =   ²log 25/²log 16

                   =   ²log 5²/²log 2⁴

                   =   2 ²log 5/4 ²log 2

                   =  

                   =  

3.      Determine the value of a given equation.

a.       ⁵log (2a – 5) = ⁵log (3a – 8)

b.      ³log (–2a – 1) –  ³log (a – 7) = 1

Solution :

a.       ⁵log (2a – 5) = ⁵log (3a – 8)                                                                                                                             2a – 5 = 3a – 8                                                                                                                                                 3a – 2a = 8 –5                                                                                                                                                                  a = 3

b.      ³log (–2a – 1) –  ³log (a – 7) =  1                                                                                                                                      ³log (–2a – 1) =  ³log (a – 7) + ³log 3                                                                                                                      ³log (–2a – 1) =  ³log ((a – 7) × 3)                                                                                                                                                           –2a – 1 = 3 (a – 7)                                                                                                                                                                 3a + 2a = 21 – 1                                                                                                                                                                                   5a = 20                                                                                                                                                                               a =  = 4

Exercise

1.      Simplify the forms of the following logarithmic.

a.       Log 25 + log 4                                     d. log 50 – log

b.      ⁶log 72 + ⁶log                                      e. ^{(x + 1)}log (x – 1) – ^{(x + 1)}log (x² – 1)

c.       ²log 3 + ²log 5                                                 f. ^alog x⁴ + 3 ^alog x – 7 ^alog x

2.      If ⁸log 5 = p, determine the value of the following logarithms.

a.       ⁴log                                                     c. ⁶⁴log 125

b.      ²log                                                 d. ⁵¹²log

3.      If ⁴log 3 = p, ⁹log 8 = q, ⁴log 6 = r, determine the value of the following logarithms.

a.       ⁴log 18                                                 c. ⁴log – ²log

b.      ²log +  64                               d. ⁸¹log 64 : ⁶⁴log 27

4.      Determine the Logarithm of a Number of More Than 10 Using the Logarithm Tables

As mentioned previously, to determine the logarithm of the number is more than 10 by using tables of logarithms, can use the properties of logarithms. At first, the number was stated earlier in the raw form a × 10ⁿ, with 1 ≤ a ≤ 10 and n are natural numbers.

Suppose b is a number that more than  10 then

            b = a × 10ⁿ                                                           (logarithm the both sides)

log b = log (a × 10ⁿ)

log b = log a + log 10ⁿ

log b = log a + n

So, obtained log a × 10ⁿ = log a + n

From the last result, log a is called the decimal (mantis / Mantissas) of the logarithm of b which can be read at the table of logarithms, while n is called the spherical (characteristic) logarithm b. Suppose that the logarithm of a number of more than 10 is 3.748 then it can be made with the following chart.

                         3                                                                                 748

In this case we have learned how to specify a logarithmic Mantissas by reading the table of logarithms (as shown in the chart above) then consider the following example. Next, consider the following example.

      Example:

Determine the value of the following logarithms.

a.       Log 542

b.      Log 350000

Solution:

a.       Log 542 = log (5.42 × 10²)

 = log 5.42 + log 10²

  = 0.734 + 2

 = 2.734

b.      Log 350000 = log (3.5 × 10⁵)

            = log 3.5 + log 10⁵

= 0.544 + 5

= 5.544

5.      Determine the Logarithm of a Number Between 0 and 1 Using Logarithm Tables

Just as the numbers that more than 10, to determine the logarithm of the number between 0 and 1 with a logarithm table, the first number's change in raw form a × 10^-n, with 1 ≤ 0 < 10 and n are natural numbers. Suppose b is a number between 0 and 1 then b = a × 10^-n (logarithm the both sides)

log b = log (a × 10^-n)

                   log b = log a + log 10^-n

                    log b = log a – n

So, obtained log (a × 10^-n) = log a – n. From the last result, log a is called the decimal (mantis / Mantissas) of the logarithm of  b which can be read at the table of logarithms, while -n is called the spherical (characteristic) logarithm b. Consider the following example.

      Example:

Determine the value of the following logarithms.

a.       Log 0.0761

b.      Log 0.003492

Solution:

a.       Log 0.0761 = log (7.61 × 10^-2)

= log 7.61 + log 10^-2

= 0.8814 + (-2)

= 0.8814 – 2 = -1.1186

b.      Log 0.003492 = log (3.492 × 10^-3)

   = log 3.492 + log 10^-3

     = 0.5428 + (-3)

   = 0.5428 – 3 = -2.4572

From the example of a, it appears that the spherical part of the logarithm of the number between 0 and 1 is -2. It can be seen in the chart below.

            Log 0.761 = 0,8814                                                                             -2