Logs to the base 10 are often call common logs, whereas logs to the base e are often call natural logs. Logs to the bases of 10 and e are now both fairly standard on most calculators. Often when taking a log, the base is arbitrary and does not need to be specified. However, at other times it is necessary and must be assumed or specified.

The Four Basic Properties of Logs

1. log_b(xy) = log_bx + log_by.
2. log_b(x/y) = log_bx - log_by.
3. log_b(xⁿ) = n log_bx.      
4. log_bx = log_ax / log_ab.     

These four basic properties all follow directly from the fact that logs are exponents. In words, the first three can be remembered as: The log of a product is equal to the sum of the logs of the factors. The log of a quotient is equal to the difference between the logs of the numerator and demoninator. The log of a power is equal to the power times the log of the base.

Additional properties, some obvious, some not so obvious are listed below for reference. Number 6 is called the reciprocal property.

1. log_b1 = 0.          
2. log_bb = 1.          
3. log_bb² = 2.         
4. log_bb^x = x.         
5. b^log_bx = x.          
6. log_ab = 1/log_ba.

FOUR BASIC PROPERTIES OF LOGS

log_b(xy)   =   log_bx + log_by.
log_b(x/y)  =   log_bx - log_by.
log_b(xⁿ)   =   n log_bx.          
log_bx      =   log_ax / log_ab.     

These four basic properties all follow directly from the fact that logs are exponents. In words, the first three can be remembered as: The log of a product is equal to the sum of the logs of the factors. The log of a quotient is equal to the difference between the logs of the numerator and demoninator. The log of a power is equal to the power times the log of the base.

Additional properties, some obvious, some not so obvious are listed below for reference. Number 6 is called the reciprocal property.

log_b1    =   0.          
log_bb    =  1.          
log_bb²  =  2.         
log_bb^x  =  x.         
b^log_bx   =  x.          
log_ab   =  1/log_ba.

yo man check this out

log is an inverse function of exponential function

suppose y=a^x be an exponential function then log_a(y) is the log function

its properties are:-

1>log_c(ab)=log_c(a)+log_c(b)

2>log_c(a/b)=log_c(a)-log_c(b)

3>log_c(a^b)=(b)log_c(a)

4>log_c^b(a)=(1/b)log_c(a)

5>on mixing above two => log_a^b(c^d)=(d/b)log_a(c)

6>log_a(y)=log_m(y)/log_m(a)

7>log_a(y)=log_xy*log_ax

8>log_a(y)=1/log_y(a)

9>x^log_a(y)=y^log_a(x)

10>log_a(x₁)>log_a(x₂) <=> x1<x2 [if 0<a<1]

                               <=> x₁>x₂[if a>1]

11>log_a(y)>x <=> y<a^x [if 0<a<1]

                               <=> x₁>x₂[if a>1]