Properties of Logarithmic Functions

A logarithmic function is basically the inverse of an exponential function.

Take a >0 and a≠1.  Then the inverse of the function f:RàR defined fy f(x)=a^x is log_ax which is called as logarithm of x with base a.

So the definition of logarithm is as follows.

log_ax = y ↔a^y = x.

Example:

2³=8 ==> log₂8 = 3.

Properties of logarithm:

(1)   log_a (mn) = log_a m + log_a n

log_a mn = x ==> mn = a^x

log_a m= y ==>m = a^y

log_a n = z ==> n = a^z
From all the equations,

a^x= a^y . a^z

a^x= a^y+z

x=y+z

log_a (mn) = log_a m + log_a n

(2)   log_a (m/n) = log_a m – log_a n

log_a m/n = x ==> m/n = a^x

log_a m= y ==>m = a^y

log_a n = z ==> n = a^z
From all the equations,

a^x= a^y / a^z

a^x= a^y-z

x=y-z

log_a (mn) = log_a m -log_a n

(3)   log_a mⁿ = n log_a m

log_a mⁿ = x ==>mⁿ = a^x

log_a m= y ==>m = a^y

From all the equations,

a^x= (a^y )ⁿ

log_a mⁿ = n log_a m

(4)    Change of base formula:

While using logarithms in solving equations, this formula is mostly used to change the bases.

log_a m = log_b m/log_b a

Let log_a m= x ==>a^x=m

Log_b m= y ==>b^y=m

log_b a= z==> b^z=a

So we get,

a^x= b^y

(b^z)^x=by

zx = y

x = y/z

log_a m = log_b m/log_b a

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