START - UP Graph each function ; identify any asymptotes, the primary locator point, and the domain and range : a. b. c.

OBJECTIVES : 1. SWBAT use the properties & the change of base formula to rewrite and evaluate logarithmic expressions. 2. SWBAT solve exponential and logarithmic equations including natural log. ESSENTIAL QUESTION : How can you use the properties of logarithms to condense and solve a logarithmic equation? HOME LEARNING : Math XL Quiz covering by 4/13 p. 289 #2-22 Even, 25, 27 and 29 Logarithmic Properties and Solving Exponential / Logarith mic Equations

Proper ties of Logari thms

Since logs and exponentials of the same base are inverse functions of each other they “ undo ” each other. Remember that : This means that: invers es “ undo ” each each other = 5 = 7

CONDENSE D EXPANDED Properties of Logarithms = = = = ( these properties are based on rules of exponents since logs = exponents )

Using the log properties, write the expression as a sum and / or difference of logs ( expand ). Use the Quotient property : When working with logs, re - write any radicals as rational exponents. Use the Product property : Use the Power property :

Using the log properties, write the expression as a single logarithm ( condense ). Use the Power Property : Use the Quotient Property : this directio n

More Properties of Logarithms This one says if you have an equation, you can take the log of both sides and the equality still holds. This one says if you have an equation and each side has a log of the same base, you know the " stuff " you are taking the logs of are equal.

( What power of 2 is 8 ? ) ( What power of 2 is 16 ? ) ( What power of 2 is 10 ? ) We know it must be more than 3 but less than 4, but we can ' t do this one in our without an equation. Change to exponenti al form. Now, we can take the log of both sides ( we ' ll use common log ) Use the Power property solve for x by dividing by log 2 use calculator to approximate Check by putting in your calcula tor ( we rounded so it won ' t be exact )

Change - of - Base Formula The base you change to can be any base…so generally we ’ ll want to change to a base so we can use our calculator. That would be either base 10 or base e. LOG “ commo n ” log base 10 LN “ natur al ” log base e If we generalize the process we just did we come up with the :

Most Calculators now have the Change - of - Base function built right in ! ALPH A TBL SET or F 2 5

Since 3 2 = 9 and 3 3 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3. Use a calculato r Use the Change - of - Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places.

Logarith mic and Exponent ial Equation s

Steps for Solvin g a Logari thmic Equati on — S. I. T. U. P. SIMPLIFY & ISOLATE : If the log is in more than one term, use log properties to condense TURN IT AROUND : Re - write the log equation in exponential form USE ALGEBRA : Solve for the variable. If x is in more than one term get x terms on one side and factor out the x PLUG IT IN TO CHECK ! to make sure your answer is “ legal ”

SIMPLIFY & ISOLATE : If the log is in more than one term, use log properties to condense use the product property of logs to “ condense ” under one log TURN IT AROUND : Re - write the log equation in exponential form

SIMPLIFY & ISOLATE : If the log is in more than one term, use log properties to condense TURN IT AROUND : Re - write the log equation in exponential form USE ALGEBRA : Solve for the variable. If x is in more than one term get x terms on one side and factor out the x

PLUG IT IN TO CHECK ! to make sure your answer is “ legal ” Remember that the domain of logs is numbers greater than 0 so we need to make sure that if we put our answers back in for x we won ’ t be trying to take the log of 0 or a negative number. Perfect! No Way ! We can ’ t take the log of - 1 or -4 so must throw this solution out

Steps for Solving an Exponential Equation If you have a u = b v and you can ’ t express a and b with the same base, take the log of both sides ( ln or log ) Use the 3 rd Property of Logs to move the exponent out in front Solve for the variable. If x is in more than one term get x terms on one side and factor out the x

If you have a u = b v and you can ’ t express a and b with the same base, take the log of both sides ( ln or log ) Use the power property to move the exponent out in front Solve for the variable. If x is in more than one term get x terms on one side and factor out the x

USE ALGEBRA & YOUR CALCULATOR making sure to enclose the numerator in parenthesis and the denominator in parenthesis

If you have an exponential equation with a base “ e ” …Just isolate e and then take the ln of both sides. Remember that e ’ s and ln ’ s are inverses and “ undo ” each other so they ’ ll cancel out and you can solve from there. SIMPLIFY & ISOLATE the “ e ” take the ln of both sides OR TURN IT AROUND !.644