PreCalc Lesson 5 6 Laws of Logarithms RememberLogs

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 – logb. N N • logb M = logb N iff M = N • logb Mk = k logb M, for any real number k Example 1: Express logb. MN 2 in terms of logb. M and logb. N 1 st: Recognize that you are taking the ‘log’ of a product (M)(N 2) So we can split that up as an addition of two separate logs!

Logb MN 2 = logb. M + logb. N 2 (Now recognize that we have a power on the number in the 2 nd log. = logb. M + 2 logb. N ! Example 2 : Express logb Logb M 3 in terms of logb. M and logb. N N M 3 = logb (M 3)1/2 = ½ (logb(M 3)) N (N) = ½ (log b. M 3 – logb. N) = ½ (3 log b. M – logb. N) = 3/2 log b. M – ½ logb. N Example 3: Simplify log 45 – 2 log 3 = log 45 – log 3 2 = 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 = ln x 1/3 + ln 4 ln y = ln (x 1/3)(4) So the only way ln y = ln 4 x 1/3 is if : y = 4 x 1/3 Example 5: Solve log 2 x + log 2(x – 2) = 3 log 2 x(x - 2) = 3 (Go to exponential form: 23 = x(x – 2) 8 = x 2 – 2 x 0 = x 2 - 2 x - 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!!!