E XPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4

E XPONENTIAL & L OG E QUATIONS In the previous sections, we covered: a) Definitions of logs and exponential functions b) Graphs of logs and exponential functions c) Properties of logs and exponential functions In this section,, we are going to study procedures for solving equations involving logs and exponential equations

E XPONENTIAL & L OG E QUATIONS In the last section, we covered two basic properties, which will be key in solving exponential and log equations. 1. One-to-One Properties 2. Inverse Properties

E XPONENTIAL & L OG E QUATIONS We can use these properties to solve simple equations:

E XPONENTIAL & L OG E QUATIONS When solving exponential equations, there are two general keys to getting the right answer: 1. Isolate the exponential expression 2. Use the 2 nd one-to-one property

E XPONENTIAL & L OG E QUATIONS Solve the following equation: Isolate the exponential expression: Apply the 2 nd one-to-one property

E XPONENTIAL & L OG E QUATIONS Solve the following equation: Isolate the exponential expression: Apply the 2 nd one-to-one property

E XPONENTIAL & L OG E QUATIONS Solve the following equation: Isolate the exponential expression: Apply the 2 nd one-to-one property

E XPONENTIAL & L OG E QUATIONS Solve the following equations: a) b) c)

E XPONENTIAL & L OG E QUATIONS Solving Equations of the Quadratic Type Two or more exponential expressions Similar procedure to what we have been doing Algebra is more complicated

E XPONENTIAL & L OG E QUATIONS Solve the following equation: Start by rewriting the equation in quadratic form. Factor the quadratic equation:

E XPONENTIAL & L OG E QUATIONS

Solve the following equation:

E XPONENTIAL AND LOGARITHMIC EQUATIONS Section 3.4

E XPONENTIAL & L OG E QUATIONS Solve the following equation:

E XPONENTIAL & L OG E QUATIONS Solve the following equation: Since these are exponential functions of a different base, start by taking the log of both sides

E XPONENTIAL & L OG E QUATIONS

So far, we have solved only exponential equations Today, we are going to study solving logarithmic equations Similar to solving exponential equations

E XPONENTIAL & L OG E QUATIONS Just as with exponential equations, there are two basic ways to solve logarithmic equations 1) Isolate the logarithmic expression and then write the equation in equivalent exponential form 2) Get a single logarithmic expression with the same base on each side of the equation; then use the one-to-one property

E XPONENTIAL & L OG E QUATIONS Solve the following equation: Isolate the log expression: Rewrite the expression in its equivalent exponential form

E XPONENTIAL & L OG E QUATIONS Solve the following equation. Get a single log expression with the same base on each side of the equation, then use the one-to-one property

E XPONENTIAL & L OG E QUATIONS Solve the following equation Isolate the log expression: Rewrite the expression in exponential form

E XPONENTIAL & L OG E QUATIONS In some problems, the answer you get may not be defined. Remember, is only defined for x > 0 Therefore, if you get an answer that would give you a negative “x”, the answer is considered an extraneous solution

E XPONENTIAL & L OG E QUATIONS Solve the following equation Isolate the log expression: Rewrite the expression in exponential form

E XPONENTIAL & L OG E QUATIONS Would either of these give us an undefined logarithm?

E XPONENTIAL & L OG E QUATIONS Solve the following equations: a) b) c)

E XPONENTIAL AND LOGARITHMIC EQUATIONS Section Applications

E XPONENTIAL & L OG E QUATIONS Solve the following equation:

E XPONENTIAL & L OG E QUATIONS Solve the following equation:

E XPONENTIAL & L OG E QUATIONS

How long would it take for an investment to double if the interest were compounded continuously at 8%? What is the formula for continuously compounding interest? If you want the investment to double, what would A be?

E XPONENTIAL & L OG E QUATIONS It will take about 8.66 years to double.

E XPONENTIAL & L OG E QUATIONS You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?

E XPONENTIAL & L OG E QUATIONS You have $50,000 to invest. You need to have $350,000 to retire in thirty years. At what continuously compounded interest rate would you need to invest to reach your goal?

E XPONENTIAL & L OG E QUATIONS For selected years from 1980 to 2000, the average salary for secondary teachers y (in thousands of dollars) for the year t can be modeled by the equation: y = ln t Where t = 10 represents During which year did the average salary for teachers reach 2.5 times its 1980 level of $16.5 thousand?

E XPONENTIAL & L OG E QUATIONS