1. Remember---Logs are ‘inverses’ of exponentials.
Pre-Calc Lesson 5-6
Laws of Logarithms
Remember---Logs are ‘inverses’ of exponentials.
Therefore all the rules of exponents will also work for
logs.
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 = logb M – logbN N
logb M = logb N iff M = N
logb Mk = k logb M, for any real number k
Example 1: Express logbMN2 in terms of logbM and logbN
1st: Recognize that you are taking the ‘log’ of a product  (M)(N2)
So we can split that up as an addition of two separate logs!

2. Example 3: Simplify log 45 – 2 log 3
Logb MN2 = logbM + logbN2
(Now recognize that we have a power on the number in the 2nd log.
= logbM + 2logbN !
Example 2 :
Express logb M3 in terms of logbM and logbN N
Logb M3 = logb (M3)1/2 = ½ (logb(M3))
N (N) (N)
= ½ (logbM3 – logbN)
= ½ (3logbM – logbN)
= 3/2 logbM – ½ logbN
Example 3: Simplify log 45 – 2 log 3
log 45 – 2 log 3 = log 45 – log 32
= log 45 – log 9
= log (45/9)
= log 5

3. Example 4: Express y in terms of x if ln y = 1/3 ln x + ln 4
So the only way ln y = ln 4x1/3 is if :
y = 4x1/3
Example 5: Solve log2x + log2(x – 2) = 3
log2x(x - 2) = 3
(Go to exponential form: = x(x – 2)
8 = x2 – 2x
0 = x2 - 2x - 8
0 = (x – 4)(x + 2)
x = 4, x = - 2
Now the domain of all log statements is (0, ф)  x ≠ - 2
so x = 4 is the only solution!!!