The properties of logarithms assume the following about the variables M, N, b, and x.

log _{b}b = 1

log _{b} 1 = 0

log _{b}b ^{x} = x

b ^{logbx} = x

log _{b} ( MN) = log _{b} ( M) + log _{b} ( N)

Note: Don't confuse with .
To find the latter, first evaluate each log separately and then do the division.

log _{b}M ^{x} = x log _{b}M

If log _{b}x = log _{b}y , then x = y.

.
This is known as the change of base formula.
Example 1
Simplify each of the following expressions.

log _{7} 7

log _{5} 1

log _{4}4 ^{3}

6 ^{log65}




Example 2
If log _{3} 5 ≈ 1.5, log _{3} 3 = 1, and log _{3} 2 ≈ 0.6, approximate the following by using the properties of logarithms.

log _{3} 10


log _{3} 25


log _{3} 1.5

log _{3} 200






Example 3
Rewrite each expression as the logarithm of a single quantity.



