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Sec 3NA AMaths E-learning 2008

Topic 2.3. Sec 3NA AMaths E-learning 2008. Exponential Equations. Instructions. Go through the following slides. Complete the worksheet given to you. Submit your work on 5 th Feb 2008 (Tue). Happy e-learning!  Best wishes, Mr Lee. Exponential Equations.

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Sec 3NA AMaths E-learning 2008

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  1. Topic 2.3 Sec 3NA AMathsE-learning 2008 Exponential Equations

  2. Instructions • Go through the following slides. • Complete the worksheet given to you. • Submit your work on 5th Feb 2008 (Tue). • Happy e-learning!  Best wishes, Mr Lee

  3. Exponential Equations • An exponential equation is an equation that contains a variable in the index(or exponent). • The simplest form is given aswhere a is the base, andx is the variable in the exponent.

  4. Exponential Equations • In addition, if b can be expressed as an, then we can write • Both sides of the result will have a as the base, and both x and n in the exponent. Hence where a ≠ -1, 0 or 1

  5. Examples Determine the value of n for the following. • 2n = 23 • 3n = 27 • 4n = 8(2n)

  6. Solving Exponential Equations • Here are the steps you should follow. • Convert each numerical value to powers of simplest base if possible.e.g. 4 = 22, 27 = 33, 125 = 53 • If there are negative powers, convert them to positive powers by inverting the fraction.e.g. ,

  7. Self-Practice (Section A) • Try Section A in your worksheet. • Please familiarise yourself with the common powers of 2 to 9, as they may be very useful in future.

  8. The Rules of Indices • Here are the rules of indices. • For a > 0 and positive integers p and q: • For a, b > 0 and any rational indices m and n:

  9. Limitations of Exponential Form • For all exponential equations of the form ax = b, these limitations should be noted. • As a summary, • a ≠ –1, 0 or 1. • If a > 0, then ax > 0. It is impossible to find a value x such that ax < 0.For example, if 3x = –9, there is no solution for x. (a = 3 so a > 0, and 3x < 0) • We will very rarely use negative bases.

  10. Solving Using Substitution • Not all exponential equations will be of the form ax = b. • Sometimes, we will have to use substitution to solve certain types of exponential equations. • When this happens, we sometimes end up with quadratic equations which we must then factorise and solve.

  11. Solving Using Substitution • For example, to solve4x+1 = 2 – 7(2x)we may use the substitution y = 2x. • This will give us 4(4x) = 2 – 7(2x) 4(22)x = 2 – 7(2x) 4(2x)2 = 2 – 7(2x) 4y2 = 2 – 7y

  12. Solving Using Substitution • Rearranging and solving, 4y2 + 7y – 2 = 0 (4y – 1)(y + 2) = 0 4y – 1 = 0 or y + 2 = 0y = ¼ or y = –2 • Don’t forget to replace y with 2x. 2x = ¼ or 2x = – 2 (rejected) 2x = 2–2Hence, x = –2. 2x = –2 is rejected because ax cannot be < 0 when a > 0.

  13. Self-Practice (Section B) • Try Section B in your worksheet. • Remember to rewrite numerical indices.e.g. 2x+2 = 22(2x) = 4(2x) • Also, remember you can write (52)x as (5x)2. • These should be done before you decide on suitable substitutions for each question.

  14. A Gentle Reminder • Hope you have completed the slides. • Complete the worksheet given to you. • Submit your work on 5th Feb 2008 (Tue). • Happy e-learning!  Best wishes, Mr Lee

  15. Thank You The End and have fun with e-learning!

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