Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

Chabot Mathematics §9.5bLogarithmic Eqns Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

MTH 55 9.5 Review § • Any QUESTIONS About • §9.5 → Exponential Equations • Any QUESTIONS About HomeWork • §9.5 → HW-47

Summary of Log Rules • Solving Logarithmic Equations Often Requires the Use of Logarithms Laws • For any positive numbers M, N, and a with a≠ 1

Typical Log-Confusion • Beware that Logs do NOT behave Algebraically. In General:

Exponent↔Logarithm Duality • Some Important Implications of the Properties of Logs & Exponents

Solving Logarithmic Equations • Equations containing logarithmic expressions are called logarithmic equations. • We discussed previously how certain logarithmic equations can be solved by writing an equivalent exponential equation

Solving Logarithmic Equations • Equations that contain terms of the form logax are called logarithmic equations: • To solve a logarithmic equation rewrite it in the equivalent exponential form:

Example  Solve Logarithmic Eqn • Solve for x: • Solution: • Since the domain of logarithmic functions is positive numbers, the tentative solution must be checked

? ? ? ? Example  Solve Logarithmic Eqn • Check x = ½ • The Solution Set: {x| x = ½}

Example  Solve Logarithmic Eqn • Solve for x: log2(6x + 5) = 4 • Solution: log2(6x + 5) = 4 6x + 5 = 24 6x + 5 = 16 6x = 11 x = 11/6 • The solution is x = 11/6. The check is left for us to do later

Use Properties of Logarithms • Often the properties for logarithms are needed to solve Log Eqns. • The goal is to first write an equivalent equation in which the variable appears in just one logarithmic expression. We then isolate that expression and solve as in the previous example

Example  Solve Logarithmic Eqn • Solve for x: logx + log(x + 9) = 1 • Solution log x + log(x + 9) = 1 log[x(x + 9)] = 1 x(x + 9) = 101 x2+ 9x = 10 x2+ 9x – 10 = 0 (x – 1)(x + 10) = 0 x – 1 = 0 or x + 10 = 0 x = 1 or x = –10

Example  Solve Logarithmic Eqn • Solve for x: logx + log(x + 9) = 1 • Check x = 1: log 1 + log(1 + 9) 0 + log(10) 0 + 1 = 1 TRUE • Check x = −10: log (–10) + log(–10 + 9) FALSE x = –10: • The logarithm of a negative number is undefined. Thus the only solution is 1

Example  Solve Logarithmic Eqn • Solve for x: log3(2x + 3) − log3(x− 1) = 2 • Soln: log3(2x + 3) – log3(x – 1) = 2 log3[(2x + 3)/(x – 1)] = 2 (2x + 3)/(x – 1) = 32 (2x + 3)/(x – 1) = 9 (2x + 3) = 9(x – 1) 2x + 3 = 9x – 9 x = 12/7 • The Solution Set: {x| x = 12/7}

Example  Solve Logarithmic Eqn • Solve: • Soln:

? ? ? ? Example  Solve Logarithmic Eqn • Solncont. • Check x = 2: 

? ? ? ? Example  Solve Logarithmic Eqn • Check x = 3  • The solution set is {x| x = 2, 3}

Example  Solve Logarithmic Eqn • Solve: • Soln Product Rule Definition of Logarithms

? Example  Solve Logarithmic Eqn • Soln a.cont. • Check x = 2:  • Logarithms are not defined for negative numbers, so x = 2 is not a solution.

? ? Example  Solve Logarithmic Eqn • Check x = 5:  • Thus The solution set is {x| x = 5}

Example  Solve Logarithmic Eqn • Soln Quotient Rule Definition of Logarithms

? ? ? Example  Solve Logarithmic Eqn • Check x = 4:  • Thus The solution set is {x| x = 4}

WhiteBoard Work • Problems From §9.5 Exercise Set • 46, 56, 62, 66, 68, 74, 98 • Exponent & Logarithm Laws.

All Done for Today SolveLog-EqnSystem

Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

ReCall Logarithmic Laws • Solving Logarithmic Equations Often Requires the Use of the Properties of Logarithms

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege