1. Aim: Logarithm Equations Course: Alg. 2 & Trig. Aim: How do we solve logarithm equations? Do Now:

2. Aim: Logarithm Equations Course: Alg. 2 & Trig. Properties of Logarithms For any positive numbers M, N, and b, b  1, Each of the following statements is true. log b MN = log b M + log b NProduct Property log b M/N = log b M – log b NQuotient Property log b M k = k log b MPower Property log (3 5) = log 3 + log 5 log (3 / 5) = log 3 – log 5 log 3 5 = 5 log 3 Note: log a (M + N) ≠ log a M + log a N Note: base must be the same

3. Aim: Logarithm Equations Course: Alg. 2 & Trig. Property of Equality for Log functions Solving Log Equations using Properties 1.If log A = log B, then A = B: Property of Equality for Log functions 2.Like regular equations use the inverse operation to simplify an equation Ex. log x - 1/3 log 8 = log 7 x = 14 log x - log 8 1/3 = log 7 Undo Power Law of Logarithms Undo Quotient Law of Logarithms 3. What you do to one side, do exactly to the other

4. Aim: Logarithm Equations Course: Alg. 2 & Trig. Solving Log Equations using Properties Ex. log 4 (x – 3) + log 4 (x + 3) = 2 log 4 [( x – 3)(x + 3)] = 2 Undo Product Law of Logarithms When only some terms are logarithmic, consolidate to one side in form log b = N and convert to exponential equation. Write in Exponential Form ( x – 3)(x + 3) = 4 2 Multiply x 2 – 9 = 16 Check: log 4 (5 – 3) + log 4 (5 + 3) = 2 log 4 (-5 – 3) + log 4 (-5 + 3) = 2 -8 Log 4 (-8) is undefined; +5 is only answer x =  5 x = 5

5. Aim: Logarithm Equations Course: Alg. 2 & Trig. Log Equation Problem log x + log(x – 3) = 1 log x(x – 3) = 1 Undo Product Law of Logarithms Write in Exponential Form x (x + 3) = 10 1 Multiply x 2 + 3x = 10 Log of negative number is undefined Put in Standard Quadratic Form x 2 + 3x – 10 = 0 x = -5x = 2 Solve for x Factor trinomial (x + 5)(x – 2) = 0

6. Aim: Logarithm Equations Course: Alg. 2 & Trig. Application Problem The pH of a substance is the concentration of hydrogen ions, [H + ], measured in moles of hydrogen per liter of substance. It is given by the formula Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2. 4.2 = log 10 1 – log 10 [H + ] 10 -4.2 = [H + ] 4.2 = 0 – log 10 [H + ] 4.2 = – log 10 [H + ] – 4.2 = log 10 [H + ] log 10 1 = 0 or 0.000063 moles of hydrogen Always check your answer

7. Aim: Logarithm Equations Course: Alg. 2 & Trig. Model Problem x2 x + 2 x = 0