Properties of Logarithms
Because the exponential function and the logarithmic function having the same base b are inverse functions of each other, it is true that
blog b x = x and log b bx = x for x > 0, b > 0 and b ¹ 1
From the second equality, we get two more equalities (for b > 0 and b ¹ 1):
(1) log b b1 = log b b = 1
(2) log b b0 = log b 1 = 0
The logarithmic function also possesses the following properties:
(1) log b xy = log b x + log b y
(2) log b (x / y) = log b x - log b y
(3) log b xr = r log b x
where x, y and b are positive real numbers, with b ¹ 1.
EX. log x 16 = 4
x4= 16
x4= 2
x = 2
EX. log 5 x = 3
53= x
x = 125
EX. log 10 10000 = x
10x= 10000
10x= 104
x = 4
EX. 5x= 17
log 5x= log 17
x log 5 = log 17
EX. log 5x - log (x - 5) = 1
log 5x - log (x - 5) = log 10
5x = 10(x - 5)
5x = 10x -50
5x = 50
x = 10
When dealing with a change in the base of a logarithmic function, the following equality can be used to facilitate the conversion:
where a, b and c are positive real numbers, where b ¹ 1 and c ¹ 1.
EX.