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Chapter 3

Chapter 3. 3-4 solving exponential and logarithmic functions. SAT Problem of the day. objectives. Solve simple and more complex exponential and logarithmic functions. Exponential and logarithmic functions.

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Chapter 3

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  1. Chapter 3 3-4 solving exponential and logarithmic functions

  2. SAT Problem of the day

  3. objectives Solve simple and more complex exponential and logarithmic functions

  4. Exponential and logarithmic functions Remember that exponential and logarithmic functions are one-to-one functions. That means that they have inverses. Also recall that when inverses are composed with each other, they inverse out and only the argument is returned. We're going to use that to our benefit to help solve logarithmic and exponential equations. Please recall the following facts: loga ax = x log 10x = x ln ex = x a loga x = x 10log x = x eln x = x

  5. Solving exponential equations Isolate the exponential expression on one side. Take the logarithm of both sides. The base for the logarithm should be the same as the base in the exponential expression. Alternatively, if you are only interested in a decimal approximation, you may take the natural log or common log of both sides (in effect using the change of base formula) Solve for the variable. Check your answer. It may be possible to get answers which don't check. Usually, the answer will involve complex numbers when this happens, because the domain of an exponential function is all reals.

  6. Example#1 Solve the equation 3 x=27.

  7. Example#2 Solve the equation 3 e 2x = 35 for x and use a calculator to give a 4 decimal place approximation answer..

  8. Example#3 Solve the equation 3 + e 3x + 2= 23 for x and use a calculator to give a 4 decimal place approximation answer.

  9. Example#4 4·52x = 64 52x = 16 log552x = log516 2x = log516 2x = 2x1.723 x0.861 Solve for x : 4·52x = 64 .Solution:

  10. Student guided practice Do problems 1-6 in your worksheet

  11. Logarithmic Equations Use properties of logarithms to combine the sum, difference, and/or constant multiples of logarithms into a single logarithm. Apply an exponential function to both sides. The base used in the exponential function should be the same as the base in the logarithmic function. Another way of performing this task is to write the logarithmic equation in exponential form. Solve for the variable. Check your answer. It may be possible to introduce extraneous solutions. Make sure that when you plug your answer back into the arguments of the logarithms in the original equation, that the arguments are all positive. Remember, you can only take the log of a positive number.

  12. Example#5 Solve for x : log3(3x) + log3(x - 2) = 2 .log3(3x) + log3(x - 2) = 2 log3(3x(x - 2)) = 2 32 = 3x(x - 2) 9 = 3x2 - 6x3x2 - 6x - 9 = 0 3(x2 - 2x - 3) = 0 3(x - 3)(x + 1) = 0 x = 3, - 1

  13. Example#5 ln (x + 4) + ln (x - 2) = ln 7 First we use property 1 of logarithms to combine the terms on the left. ln (x + 4)(x - 2) = ln 7 Now apply the exponential function to both sides. eln (x + 4)(x - 2) = eln 7 The logarithmic identity 2 allows us to simplify both sides. (x + 4)(x - 2) = 7 x2 + 2x - 8 = 7 x2 + 2x - 15 = 0 (x - 3)(x + 5) = 0 x = 3 or x = -5 x = 3 checks, for ln 7 + ln 1 = ln 7.

  14. Example#6 solve −6log 3 (x− 3) = −24 for x

  15. Example#7 log 9 (x+ 6) − log 9x= log 92

  16. Do problems Do problems 1-8 in your worksheet

  17. Video Let’s watch solving exponential and logarithmic function

  18. Homework Do problems 35 -42 in your book page 217

  19. closure Today we learned to solve exponential and logarithmic functions Next class we are going to learn about exponential and logarithmic models

  20. Have a nice day

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