Logarithm Definition Types and Core Concepts

Logarithm Meaning

A logarithm is a word and concept coined by John Napier, a Scottish mathematician. A logarithm is derived from the combination of two Greek words that are logos that means principle or thought and arithmos means a number.

Logarithm Definition

A logarithm is the power to which must be raised to get a certain number. It is denoted by the log of a number. Example: log(x). 

Logarithm Examples for class 9, 10, and 11;

if y=ax

then, logay= x

a is the base.

x is the exponent.

where,  a>0, a≠1, y≠0

For example: 25= 32 

Log232= 5

Types of Logarithms

• Common Logarithm: 

The logarithm whose base is 10 is called the Common logarithm. It is basically how many 10 must be multiplied to get a given number. It is denoted as log 10 or log.

For example: 

log 100

= log 10² 

= 2log 10

= 2 

• Natural Logarithm: 

The logarithm whose base is considered e (Euler's constant) is called Natural logarithm. It is the number of e that must be multiplied to get the given number. 

For Example:

 e= 2.71828

The natural log of a number for e.g. 56 is denoted by Ln56.

In 56= 4.02535169

Properties of Logarithm 

1. logb (mn) = n logb m  (Log of power rule)

Example: log10 (23)= 3log102

2. logb (mn)= logb m + logbn  (Product rule)

Example: log10 (14)= log10 (7*2)= log10 7 + logb 2

3. logb (m/n)= logb m – logb n  (Quotient rule) 

Example: log10 (4/5)= logb 4 – logb 5

4. logb 1= 0

5. logb (m) = ln m / ln b (Change of base rule)

or logb (n) = log10n / log10 b

For example: log10 (7) = ln 7 / ln 10

6. ln(1/x) = -ln (x) (Log of reciprocal rule)

For example: .  ln(1/5) = -ln (5)

Note for Logarithm Class 9

1. logb (m+n) ≠ logb m + logb n

2. logb (m-n) ≠ logb m - logb n

Applications of Logarithms

In this technological era, people are always finding ways to do things in simpler and easier ways. Therefore, people invented calculators and logarithms to make mathematical equations easier to get solved. 

So, the advantages of understanding the concept of Logarithm:

• In many scientific research and studies, a logarithm is used.

• Logarithms help to find the pH value in chemistry because the value for pH can be small, so we use the logarithm to have a range for using it for small numbers.

• Logarithms are widely used in banking.

• Logarithms are used to find the half-life of radioactive material.

• It is used to find out the seismic waves. 

• It plays a very crucial role in the field of medicine or engineering.

Some Solved Logarithm Problems for Class 10

1. If loga p= q, Express aq-1 in Terms of a and p.

Solution: 

loga p= q

ap= q

ap/a = q/a

aq-1 = p/a

2. Solve: (3+log7 x)/(4 – 2 log7x) = 2

Solution: 

3+log x = 8 – 4log7x

5log7x  = 5

log7x = 1

x = 71 = 7

3. Solve (log 2x) 2  – log2x2 - 32 = 0. Given x is an Integer.

Solution: 

(log2x)2 – log 2x4 - 32 = 0.  

⇒ (log 2x)2 – 4log2 x - 32 = 0......(1)

log 2x = y (say)

(i) ⇒ y2 – 4y – 32 = 0

⇒ y2 – 8y + 4y – 32 = 0

⇒ y (y – 8) + 4 (y – 8) = 0

⇒ (y – 8) (y + 4) = 0

⇒ y = 8, -4

⇒ log2x = 8 or log2x = - 4

X=28 = 256.

Since, x is an integer therefore, x = 256.

4. Express [\[\frac{1}{3}\]]\[^{4}\] = \[\frac{1}{81}\] in Logarithmic Form.

Answer: 

Take the log of base \[\frac{1}{3}\] on both sides

[\[\frac{1}{3}\]]\[^{4}\] = \[\frac{1}{81}\] 

⇒ log\[_{\frac{1}{3}}\] [\[\frac{1}{3}\]]\[^{4}\] = log\[_{\frac{1}{3}}\] \[\frac{1}{81}\]

⇒ 4log\[_{\frac{1}{3}}\] \[\frac{1}{3}\] = log\[_{\frac{1}{3}}\] \[\frac{1}{81}\]         (Since log\[_{b}\] a\[^{n}\] = n log\[_{b}\] a)    

⇒ 4 = log\[_{\frac{1}{3}}\] \[\frac{1}{81}\]       (Since log\[_{b}\] b = 1)

⇒ log\[_{\frac{1}{3}}\] \[\frac{1}{81}\] = 4

Hence, the logarithmic form is,