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Chapter 11 Sec 4

Chapter 11 Sec 4. Logarithmic Functions. Graph an Exponential Function. If y = 2 x we see exponential growth meaning as x slowly increases y grows rapidly. The inverse of this function is x = 2 y this represent quantities that increase or decrease slowly.

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Chapter 11 Sec 4

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  1. Chapter 11 Sec 4 Logarithmic Functions

  2. Graph an Exponential Function If y = 2xwe see exponential growth meaning as x slowly increases y grows rapidly. The inverse of this function is x = 2y this represent quantities that increase or decrease slowly. In general the inverse of y = bx is x = by. x = by y is called the logarithmof x and is usually written as y = logbxand is read log base b of x. 6 5 4 3 2 1 -3 -2 -1 1 2 3 4

  3. Logarithm with Base b

  4. Logarithmic to Exponential Form Write each expression in exponential form. logb N = k if and only if bk= N a. log8 1 = 0 b = 8 N = 1 k = 0 b. log5 125 = 3 b = 5 N = 125 k = 3 c. log13 169 = 2 b = 13 N = 169 k =2 b = 2 N = 1/16 k =-4 80 = 1 53 = 125 132 = 169

  5. Exponential to Logarithmic Form Write each expression in logarithmic form. logb N = k if and only if bk= N a. 103 = 1000 b = 10 N = 1000 k = 3 b. 33 = 27 b = 3 N = 27 k = 3 b = 1/3 N = 9 k = - 2 b = 9 N = 3 k =1/2 log10 1000 = 3 log3 27 = 3

  6. Evaluate Logarithmic Expressions Evaluate log2 64, remember logb N = k and bk = N so..find k a. log3 243 3k = 243 3k = 35 so… k = 5 Now, log3 243 = 5 a. log2 64 2k = 64 2k = 26 so… k = 6 Now, log2 64 = 6 = k = k

  7. Evaluate Logarithmic Expressions Evaluate each expression. logb N = k and bk = N a. log6 68 log6 68 = k 6k = 68 so… k = 8 log6 68 = 8 b =3 k = log3 (4x - 1) log3 N = log3 (4x - 1) so… N = 4x -1

  8. Properties

  9. Example Solve each equation X

  10. Chapter 11 Sec 5 Common Logarithm

  11. Common Logs • Common Logarithms are all logarithms that have a base of 10…log10 x = log 3 • Most calculators have a key for evaluation common logarithms. LOG Example 1. Use a calculator to evaluate each expression to four decimal places. a. log 3 b. log 0.2 .4771 3 LOG ENTER –.6990 0.2 LOG ENTER

  12. Solving Solve 3x = 11 3x = 11 log 3x = log11 x log 3 = log 11 Solve 5x = 62 5x = 62 log 5x = log62 x log 5 = log 62 Equality property Power property Divide each side by log 3

  13. Change of Base Formula • This allows you to write equivalent logarithmic expressions that have different bases. For example change base 3 into base 10

  14. Change of Base Express log 4 25 in terms of common logarithms. Then approximate its value.

  15. Antilogarithm • Sometime the logarithm of x is know to have a value of a, but x is not known. • Then x is called the antilogarithm of a, written as antilog a. • So, if log x = a, then x = antilog a. • Remember that the inverse (or antilog) of a logarithmic function is an exponential function. • ie log x = 2.7 → x = antilog 2.7 or 102.7 • x =501.2

  16. Daily Assignment • Chapter 11 Sections 4 & 5 • Text Book • Pgs 723 – 724 • #21 – 51 Odd; • Pgs 730 – 731 • #19 – 45 Odd;

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