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,

a^x = n ⇔ log_a 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 log_e. 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

👉 log_e = ln

For example:

e^x = 3 ⇒ log_e 3 = x or ln 3 = x

e^x = 5 ⇒ log_e 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. log_a 1 = 0
2. log_a a = 1
3. log_a (mn) = log_a m + log_a n  (Product property)
4. log_a \((\frac {m}{n})\) = log_a m - log_a n  (Division property)
5. log_a (m)ⁿ = n log_a m  (Exponential property)
6. log_a^k (m) = \(\frac {1}{k}\) log_a m
7. log_a^k (m)ⁿ = \(\frac {n}{k}\) log_a m
8. log_m n = \(\frac {log_a(n)}{log_a(m)}\)   (Base changing property)
9. log_m n = \(\frac {1}{log_n(m)}\)   (Switch base property)
10. a^{log_a (n)} = n 
11. log_a (m + n) = log_a m + log_a (1 + nm)
12. log_a (m - n) = log_a m + log_a (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  log_x 4 + log_x 16 + log_x 64 = 12 than what is the value of x?   

Solution:

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

⇒ 2 log_x 2 + 4 log_x 2 + 6 log_x 2 = 12

⇒ 12 log_x 2 = 12

⇒ log_x 2 = \(\frac {12}{12}\)

⇒ log_x 2 = 1

⇒ x¹  = 2  (∵ a^x = n ⇔ log_a n = x)

⇒ x = 2

Qus3- (25)^{log₆₂₅ (36)}

Solution:

⇒ (25)^{log₂₅₂ (36)}   

⇒ (25)^{½log₂₅ (36)}    [∵ log_a^k (m) = \(\frac {1}{k}\) log_a m]

⇒ (25)^{log₂₅ (36)½}  [∵ a^log_a (n) = n]

⇒ (36)^½

⇒ 6

Exponential Series (Formula)

General form

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

Other series

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

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

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

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

e^x - 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

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

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

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

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

log_e (1 + x) - log_e (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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