Logarithm is one of basic mathematics concept. Logarithm counts repeated same number in multiplication form. Operating logarithm means determine how many repeated numbers (*b*) to get the result (*x*).

Logarithm is used to many sectors in life. Some of them are calculate the magnitude of earthquake using Richer Scale, loudness of sound in decibels (dB), acidity, etc..

Its general form is

**log**_{b}**(x)** or **log**_{b}**x**

*b* as the base.

*x* as the result of multiplying a same number.

Logarithm base is 10. Consequently, if there is “logx” or “log(x)” (there is no base) then it means the base is 10.

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## Special Forms of Logarithm

There are special forms of logarithm:

**Common logarithm**

Common logarithm has 10 as the base. The form is **log(x)**. automatically it means log_{10}(x)

**Natural logarithm**

The base of natural logarithm is *e* (Euler’s number:2.71828182…). The form is log_{e}(x) and it can be written as **ln(x)**.

**Natural exponential function**

Natural exponential function is **e**** ^{x}**.

**Exponents and Logarithms**

Exponent has close relation with logarithm. Exponent shows how many times a number in multiplication form. It is also can be written as logarithm. In another words, logarithm is another form of exponent or it is also called as invers form of exponent.

2 x 2 x 2 x 2 x 2 = 2^{5} = 32

It can be written as logarithm.

log_{2}(32) = 5

in another word,

2^{5} = 32 « log_{2}(32) = 5

**Properties of Logarithm**

There are some properties of logarithm. It is also known as rules of logarithm. It is used to operate logarithm to solve the problems. The basic of properties of logarithm are simplifying and expanding logarithm. There is also other properties of logarithm.

**1. Simplifying Logarithm**

Simplifying means make it simpler. It is also called condensing logarithm. Simplifying logarithm consist of three basic concepts, they are addition rule, subtraction rule, and power rule of logarithm.

**Addition rule**

**log**_{a}**(m) + log**_{a}**(n) = log**_{a}**(m.n)**

log_{2}(2) + log_{2}(3) = log_{2}(2.3)

= log_{2}(6)

**Subtraction rule**

**log**_{a}**(m) – log**_{a}**(n) = log**_{a}**(m/n)**

log(8) – log(2) = log(8/2)

= log(4)

**Power rule (multiplication by scalar)**

**b.log**_{a}**(m) = log**_{a}**(m**^{b}**) **

3. log_{2}(2) = log_{2}(2^{3})

= log_{2}(8)

**2. Expanding Logarithms**

Expand means change the form based on each parts or components. Expanding logarithm consist of four basic concepts, they are zero-exponent, product, quotient, and power rules.

**Zero-exponent rule**

**Log**_{a}**1 = 0**

In exponent form it will be a^{0} =1.

Log_{3}1 = 0 « 3^{0} = 1

**log**_{a}**a = 1 ****«**** a**^{1}** = a**

log_{3}3 = 1 « 3^{1} = 3

**Product rule**

**log**_{a}**(m.n) = log**_{a}**(m) + log**_{a}**(n)**

log_{2}(2.3) = log_{2}(2) + log_{2}(3)

= 1 + log_{2}(3)

**Quotient rule**

**log**_{a}**(m/n) = log**_{a}**(m) – log**_{a}**(n)**

log(8/2) = log(8) – log(2)

**Power rule**

**log**_{a}**(m**^{b}**) = b.log**_{a}**(m)**

log_{2}(2^{3}) = 3. log_{2}(2)

**Other properties of logarithm****Log**_{a}^{b}**(m) = (1/b).log**_{a}**(m)**

Log_{9}(27) = log(_{3}^{2})(27)

= ½ . log_{3}(27)

= ½ . 3 → log_{3}(27) = log_{3}(3^{3}) = 3. Log_{3}(3) = 3.1 = 3

= 3/2

**log**_{a}**m = log(m) / log(a)**

log_{3}(5) = log(5) / log(3)

**log**_{a}**(b).log**_{b}**(c).log**_{c}**(d) = log**_{a}**(d)**

log_{2}(3).log_{3}(4).log_{4}(8) = log_{2}(8)

= log_{2}(23)

= 3.log_{2}(2)

= 3

**log**_{a}(b) = 1 / log_{b}(a)

log_{3}(7) = 1 / log_{7}(3)

**ln(e**^{x}**) = x**

ln(e^{2}) = 2

**Example**

1. Determine the result and write in logarithm form

a. 3^{2} = …

b. 10^{5} = …

c. 2^{-4} = …

a. 3^{2} = 9 « log_{3}(9) = 2

b. 10^{5} = 100.000 « log(100.000) = 5

c. 2^{-4} = ( ½ )^{4} = 1/16 = 0.0625 « log_{2}(0.0625) = -4

2. Log_{2}(24) + log_{2}(20) – log_{2}(15) = …

Log_{2}(24) + log_{2}(20) – log_{2}(15) = log_{2}[(24×20)/15]

= log_{2}(32)

= 5

3. If log_{3}(5) = y then log_{9}(625) + log_{3}(15) = …

If log_{3}(5) = y then

log_{9}(625) + log_{3}(15)

= log_{3}^{2}(5^{4}) + log_{3}(5 × 3)

= (4/2).log_{3}(5) + log_{3}(5) + log_{3}(3)

= 2. log_{3}(5) + log_{3}(5) + log_{3}(3)

= 2y + y + 1

= 3y + 1

4.