# Logarithms (exponential and logarithmic series)

## Logarithms formula, properties and rules

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### Introduction (What is a logarithm?)

The logarithm is another way of writing an exponential equation. It was introduced by Scottish mathematician John Napier in 1614. We know that numbers in exponential form write like that,

ax = n ⇔ loga n = x

where a > 0, n > 0, a ≠ 1

For Example:

24 = 16

It can write in logarithm form like that,

log2 16 = 4

### Types of Logarithm

Mainly two types of logarithms we use,

1. Natural Logarithm
2. Common Logarithm

#### Natural Logarithm

The Logarithm with base "e" is called Natural Logarithm. A natural logarithm writes like loge. Here "e" is an Euler's number. Its value is approximately equal to 2.71828 and first, it is introduced by swiss mathematician Leonhard Euler (Pronounce as "Oiler") in 1731. But many times it is represented as ln

👉 loge = ln

For example:

ex = 3 ⇒ loge 3 = x or ln 3 = x

ex = 5 ⇒ loge 5 = x or ln 5 = x

Common Logarithm

The Logarithm with base "10" is called common logarithm. The common logarithm is represented as log10. The common logarithm describes how many times we multiple the number '10' to get the required value or number.

For example:

log10 1000 = 3 ⇒ 103 = 1000 ⇒ 10 × 10 × 10 = 1000

This describes we multiply 10 three times to get the value '1000'.

### Logarithmic Properties, Rules and Formulas

In logarithm questions, many properties are used that are given below 👇

1. loga 1 = 0
2. loga a = 1
3. loga (mn) = loga m + loga n  (Product property)
4. loga $$(\frac {m}{n})$$ = loga m - loga n  (Division property)
5. loga (m)n = n loga m  (Exponential property)
6. logak (m) = $$\frac {1}{k}$$ loga m
7. logak (m)n = $$\frac {n}{k}$$ loga m
8. logm n = $$\frac {log_a(n)}{log_a(m)}$$   (Base changing property)
9. logm n = $$\frac {1}{log_n(m)}$$   (Switch base property)
10. aloga (n) = n
11. loga (m + n) = loga m + loga (1 + nm)
12. loga (m - n) = loga m + loga (1 - $$\frac {n}{m}$$)

### Some examples of logarithms

Qus1- log3 (729)

Solution:

⇒ log3 (3)6

⇒ 6 log3 3

⇒ 6 × 1

⇒ 6

Qus2- log10 (0.1)

Solution:

⇒ log10 $$(\frac {1}{10})$$

⇒ log10 1 - log10 10

⇒ -1

Qus3- If  logx 4 + logx 16 + logx 64 = 12 than what is the value of x?

Solution:

⇒ logx (2)2 + logx (2)4 + logx (2)6 = 12

⇒ 2 logx 2 + 4 logx 2 + 6 logx 2 = 12

⇒ 12 logx 2 = 12

⇒ logx 2 = $$\frac {12}{12}$$

⇒ logx 2 = 1

⇒ x1  = 2  (∵ ax = n ⇔ loga n = x)

⇒ x = 2

Qus3- (25)log625 (36)

Solution:

⇒ (25)log252 (36)

⇒ (25)½log25 (36)    [∵ logak (m) = $$\frac {1}{k}$$ loga m]

⇒ (25)log25 (36)½  [∵ aloga (n) = n]

⇒ (36)½

⇒ 6

### Exponential Series (Formula)

#### General form

ax = $$1+\frac{x\space log_ea}{1!}+\frac{(x\space log_ea)^2}{2!}+\frac{(x\space log_ea)^3}{3!}+ .......∞$$

Other series

ex = $$1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+ .....+\frac{x^n}{n!}+....∞$$

ex = $$1-\frac{x}{1!}+\frac{x^2}{2!}-\frac{x^3}{3!}+ .....+(-1)^n \frac{x^n}{n!}+....∞$$

eax = $$1+\frac{ax}{1!}+\frac{(ax)^2}{2!}+\frac{(ax)^3}{3!}+ .....+\frac{(ax)^n}{n!}+....∞$$

ex + e-x = $$2\left [1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^4}{4!}+ .......∞ \right ]$$

ex - e-x = $$2\left [\frac{x}{1!}+\frac{x^3}{3!}+\frac{x^5}{5!}+ .......∞ \right ]$$

e = $$1+\frac{1}{1!}+\frac{1}{2!}+\frac{3}{3!}+ .......∞$$

e-1 = $$1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}+.......∞$$

### Logarithmic Series

loge (1 + x) = $$x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+ ......∞$$

loge (1 - x) = $$-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}+ ......∞$$

loge (1 + x) + loge (1 - x) = $$-2\left [\frac{x^2}{2}+\frac{x^4}{4}+\frac{x^6}{6}+ .......∞ \right ]$$

loge (1 - x2) = $$-2\left [\frac{x^2}{2}+\frac{x^4}{4}+\frac{x^6}{6}+ .......∞ \right ]$$

loge (1 + x) - loge (1 - x) = $$2\left [x+\frac{x^3}{3}+\frac{x^5}{5}+ .......∞ \right ]$$

$$log_e \frac {(1 + x)}{(1 - x)} = 2\left [x+\frac{x^3}{3}+\frac{x^5}{5}+ .......∞ \right ]$$

If more questions related to logarithms, exponential and logarithmic series then comment below.👇

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