1. LOGARITHMIC EQUATIONS AND IDENTITIES Patterns#12

2. Prerequisites 1) a) Evaluate the following. i) log 100ii) log 10iii) log 1 iv) log 0v) log(-10) 2) Write an expression whose value is 2, using each logarithm. a) base 3b) base 5c) base 10 3) Solve by factoring. a) x 2 – x – 20 = 0b) x 2 – 4x = 0 c) x(x – 5) = 24

3. Solutions to Prerequisites 1) a) i) 2ii) 1iii) 0iv, v) undefined 2) a) log 3 9b) log 5 25c) log 100 3) a) -4, 5b) 0, 4c) -3, 8

4. 1) Solve each equation and check. a) log 3 (x – 4) + log 3 x = log 3 21 b) log 2 (x + 3) + log 2 (x + 5) = 3

5. Solutions 1) a) log 3 (x – 4) + log 3 x = log 3 21 log 3 ((x – 4)x) = log 3 21 log 3 (x 2 – 4x) = log 3 21 x 2 – 4x = 21 x 2 - 4x – 21 = 0 (x – 7)(x + 3) = 0 x = 7, -3 *but -3 is extraneous so x = 7

6. Solutions 1) b)log 2 (x + 3) + log 2 (x + 5) = 3 log 2 ((x + 3)(x + 5)) = 3 log 2 (x 2 + 8x + 15) = 3 2 3 = x 2 + 8x + 15 8 = x 2 + 8x + 15 0 = x 2 + 8x + 7 0 = (x + 7)(x + 1) x = -7, -1*but -7 is extraneous* so x = -1

7. Consider the identity log a 5 + x = log a (5a x ) a) Verify the identity when a = 10 and x = 2 b) Determine the values of x for which each side of the identity is defined c) Verify the identity graphically d) Prove the identity for any positive base a and any value of x

8. Solutions a) Both sides are approximately 2.698 97 b) x is defined for all reals c) graph: Y1=log(5)+x Y2=log(5*10^x) you get the same line for both

9. Solutions continued d)log a 5 + x = log a (5a x ) LS:=log a 5 + log a a x =log a (5a x ) =RS

10. Textwork p.152/1 - 9