5-6

5-6 Laws of Logarithms

In this lesson, we learned how to prove and use laws of logarithms.

-If M and N are positive real numbers and b is a positive number other than 1, then:

1. logb MN = logb M + logb N
2. logb M/N = logb M - logb N
3. logb M = logb N if and only M = N
4. logb MO^k = klogb M, for any real number k

Example 1: Write the expression in terms of log M and log N.

a) log (MN)^2

= 2 (logM + logN)

b) log 1/M

= -log M

Example 2: Write the expression as a rational number or as a single logarithm.

a) log 8 + log 5 - log 4

= log 8(5)/4

=log 10 (There is no base, so it's automatically a base of 10, which makes the bases the same. - Therefore, the bases cancel out)

= 1

b) 2 ln6 - ln 3

= ln 36 - ln 3

= ln 12