Logarithmic and Exponential Form
Any number can be written in Logarithmic and Exponential Form

Logarithmic Form
log_bx = a

Exponential Form
x = b^a

Logarithmic Function and Exponential function are inverse function to each other
A function which is written in exponential form can be written into logarithmic form given both have the same base.

y = log₃x

is equivalent to

3^y = x

Let take a number 3⁴

3⁴ = 3 x 3 x 3 x 3 = 81

log₃81 = 4

Here
3 is Base
4 is exponent

What is the exponent of the below statement? What is the value of x?

log₅125 = x
5 x 5 x 5 = 125
x = 3

log₄1 = x
x = 0

Why Know Logarithms?
Let us take the set of below numbers

1
10
1000
100000(10⁵)
100000000(10⁸)
100000000000000(10¹⁴ )

When we try to plot the numbers we won’t be able to accommodate the small and the higher values since the variations are huge and the difference between the first and last number is huge. In such a case, we can represent the numbers in terms of logarithms with respect to some other decimal value. Now the above numbers could be written as below.

log₁₀1 = 0
log₁₀1 = 1
log₁₀1000 = 3
log₁₀x = 5
log₁₀x = 8
log₁₀x = 14

Now it is easy to accommodate 0,1,3,5,8,14 in the graph.

Practical application of Logarithms?
The best practical usage of the logarithm is the Richter scale which is used to measure the earthquake. Richter scale is a logarithmic scale with base 10. Let’s say we there are earthquakes in 3 locations as below
India – 6.0
Thailand – 7.0
Japan – 9.0

Now the difference of intensity between the earthquake in India and Thailand is 10 Times Stronger
The difference of intensity between the earthquake in India and Japan is 1000 Times Stronger

Fundamental properties

1st Property
log_bx = a
x = b^a
log_bx = b^a

2nd Property
log_bb = x
logb^x = b
x = 1

log4⁴ = 1
log25²⁵ = 1

3rd Property
log_bbⁿ = x
b^x = bⁿ
x = n

log₁₁11³ = 3
log₄4⁵ = 5

1. log_bx = b^a
2. log_bb = 1
3. log_bbⁿ = n

Condition

log_ba – the value of a(argument) should be greater than 0 – Incorrect

log_b0 is undefined and log_b(-ve no) is incorrect – Incorrect

log_b1= 0 is correct

log_b(1/2)= -1 is correct

log₁a is incorrect- base cannot be 1

log_(-ve)a is incorrect- base cannot be -ve number

Change of Base Rule

log_ba = log_xa/log_xb where x>0, x != 1 and a,b >0

log denotes log to the base 10 and ln or log _e represents natural log.

e – Maximum possible result after continuously compounding 100% growth over a time period

log_ab x log_ba = 1

Logarithm Rules

1. log_b(xy)= log_bx + log_by
2. log_b(x/y)= log_bx – log_by
3. log_baⁿ = nlog_ba