Pre-Calc Lesson 5-6

Laws of Logarithms

Remember---Logs are ‘inverses’ of exponentials. Therefore all the rules of exponents will also work forlogs.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 N2. 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 logbN1st: 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!

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

Example 4: Express y in terms of x if ln y = 1/3 ln x + ln 4 ln y = 1/3 ln x + ln 4 ln y = ln x1/3 + ln 4 ln y = ln (x1/3)(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: 23 = x(x – 2) 8 = x2 – 2x 0 = x2 - 2x - 8 0 = (x – 4)(x + 2) x = 4, x = - 2Now the domain of all log statements is (0, ф) x ≠ - 2 so x = 4 is the only solution!!!