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Understanding Logarithmic Functions

Learn the basics of logarithmic functions, the relationship with exponential functions, and how to convert between exponential and logarithmic forms. Practice evaluating logarithmic expressions and solving logarithmic equations. Includes common logarithms and antilogarithm concepts.

sylvester-evans
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Understanding Logarithmic Functions

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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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