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# Common and Natural Logarithms – Explanation & Examples

The **logarithm of a number** is the power or exponent by which another value must be raised in order to produce an equivalent value of the given number.

The concept of logarithms was introduced in the early 17th century by John Napier – a Scottish mathematician. Later, scientists, navigators and engineers adopted the concept to perform computation using logarithmic tables.

The logarithm of a number is expressed in the form of;

log_{ b }N = x, where b is the base and can be any number except 1 and zero; x and N are the exponent and argument respectively.

**For example**, the logarithm of 32 to base 2 is 5 and can be represented as;

log _{2 }32 = 5

Having learned about logarithms, we can note that, the base of a logarithmic function can be any number except 1 and zero. However, there are other **two special types of logarithms** that are frequently used in mathematics. These are common logarithm and natural logarithm.

## What is a Common Logarithm?

**A common logarithm has a fixed base of 10.** The common log of a number N is expressed as;

log _{10 }N or log N. Common logarithms are also known as decadic logarithm and decimal logarithm.

If log N = x, then this logarithmic form can be represented in exponential form as well i.e. 10 ^{x} = N.

Common logarithms have a wide application in science and engineering. These logarithms are also called Briggsian logarithms because in 18^{th} century, a British mathematician Henry Briggs introduced it. For example, the acidity and alkalinity of a substance is expressed in exponential.

The **Richter scale** for measurement of earthquakes and the decibel for sound are usually expressed in logarithmic form. It is so common that if you find no base written, you can assume it to be log x or common log.

The **basic properties of common logarithms** are same as the properties of all logarithms.

These include product rule, quotient rule, power rule, and zero exponent rule.

**Product rule**

The product of two common logarithms is equal to the sum of individual common logarithms.

⟹ log (m n) = log m + log n.

**Quotient rule**

The division rule of common logarithms states that the quotient of two common logarithmic values is equal to the difference of each common logarithm.

⟹ log(m/n) = log m – log n

**Power rule**

The common logarithm of a number with an exponent is equal to the product of exponent and its common logarithm.

⟹ log (m ^{n}) = n log m

**Zero Exponent Rule**

⟹ log 1 = 0

## What is a Natural Logarithm?

**The natural logarithm of a number N is the power or exponent to which ‘e’ has to be raised to be equal to N. The constant ‘e’ is the Napier constant and is approximately equal to 2.718281828.**

ln N = x, which is the same as N = e ^{x}.

**Natural logarithm** is mostly used in pure mathematics such as calculus.

The basic properties of natural logarithms are same as the properties of all logarithms.

**Product Rule**

⟹ ln (ab) = ln (a) + ln (b)

- Quotient Rule

⟹ ln (a/b) = ln (a) – ln (b)

**Reciprocal Rule**

⟹ ln (1/a) = −ln (a)

**Power Rule**

⟹ ln (a ^{b}) = b ln (a)

*Other properties of natural log are:*

- e
^{ln (x) }= x - ln (e
^{ x}) = x - ln (e) = 1
- ln (∞) = ∞
- ln (1) = 0

Scientific and graphing calculators have keys for both common and natural logarithms. The key for natural log is labelled “*e”* or “ln” while that of the common logarithm is labelled “log”.

Now, let’s check our understanding of the lesson by attempting a few problems of natural and common logarithms.

*Example 1*

Solve for x if, 6 ^{x}^{ + 2} = 21

__Solution__

Express both sides in common logarithm

log 6 ^{x}^{ + 2} = log 21

Applying the power rule of logarithms, we get;

(*x* + 2) log 6 = log 21

Divide both sides by log 6.

x + 2 = log 21/log 6

x + 2 = 0 .5440

x = 0.5440 – 2

x = -1.4559

*Example 2*

Solve for x in e^{2x} = 9

__Solution__

ln e^{3x} = ln 9

3*x* ln e = ln 9

3*x* = ln 9

isolate x by dividing both sides by 3.

x = 1/3ln 9

x = 0. 732

*Example 3*

Solve for x in log 0.0001 = x

__Solution__

Rewrite the common log. in exponential form.

10^{x }= 0.0001

But 0.0001 = 1/10000 = 10^{-4}

Therefore,

x = -4

*Practice Questions*

1. Find x in each of the following:

a. ln x = 2.7

b. ln (x + 1) = 1.86

c. x = e ^{8} ÷ e ^{7.6}

d. 27 = e ^{x}

e. 12 = e ^{-2x}

2. Solve 2 log 5 + log 8 – log 2

3. Write log 100000 in exponential form.

4. Find the value x if log x = 1/5.

5. Solve for y if e ^{y }= (e ^{2y }) (e ^{ln 2x}).

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