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Warm up What is the inverse of the exponential function π¦ = π₯ 2 ? Solution β’ We can see the inverse is π₯ = 2π¦ by switching the variables, but how do we solve for y? β’ It turns out, we need a special function called a Logarithmic function to do this. Why do we care? β’ Logarithms are functions that used to be very helpful, but most of their value has become obsolete now that we have fancy calculators. β’ BUT! They still are extremely important in solving exponential equations. β’ That is what we will use them for. First we have to understand what a logarithm is before we can use it. Definition of a Logarithm β’ Does the π = π π₯ look familiar? Letβs use this definition to solve the warm-up. β’ We have π₯ = 2π¦ , and we want to solve for y. β’ We are going to rewrite this function using logs to solve for y. β’ π₯ = 2π¦ becomes log 2 π₯ = π¦ β’ So π β1 π₯ = log 2 π₯ What are all of these numbers? Number Exponent π¦ = log 2 π₯ Base (if no base, then base is 10) Example 1 β’ Rewrite the logarithmic equation in exponential form using the definition. Solve for y. log 1000 = π¦ Solution β’ β’ β’ β’ log 1000 = π¦ Original 1000 = 10π¦ Using definition 103 = 10π¦ 1000=103 y=3 Equal base rule. You try log 5 625 = π¦ Look at the definition, and plug in numbers where they belong. Solution β’ β’ β’ β’ log 5 625 = π¦ 625 = 5π¦ 54 = 5 π¦ y=4 Original Using Definition Get common base Common base rule One more, a little harder: Solve for x 4 log 9 9 = π₯ Solution 4 β’ log 9 9 = π₯ β’ 4 9 = 9x 1 4 β’ 9 = 9π₯ β’ x= 1 4 Original Definition of log Write root as a radical Common base rule. We will also want to find exact values of logs. β’ If the log is base 10, then it is easy to do on the calculator. If it is not base 10, we will have to put it in terms of base 10 to use the calculator. Example β’ Find the exact value of log 20 πππ log 1 β’ On your calculators, find the log button (middle left). Enter log(20). β’ You should get 1.3. β’ What do you get for log(1)? But, there is not a button if the base is not 10β¦ We will use this rule to solve for exact values of logs on our calculators. Example πππ243 log 3 243 = =5 πππ3 *To get 5, simply enter the log fraction in the calculator. Example β’ Solve 4π₯ = 128. Solution β’ 4π₯ = 128. β’ x=log 4 128 β’ πππ128 x= πππ4 β’ X=3.5 Original Rewrite using logs Use change of base rule Enter in calculator. You try β’ 6π₯ = 12 Answer β’ 6π₯ = 12 β’ π₯ = log 6 12 β’ π₯= πππ12 πππ6 β’ x=1.39 Homework β’ 5.6 Worksheet 1:all, 2:a-e, 3:Skip, 4:skip, 5:a-g