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MTH108 Business Math I

MTH108 Business Math I. Lecture 17. Chapter 7. Exponential and Logarithmic Functions. Review. Properties of exponents and radicals Exponential functions Special classes of exponential functions Cases of exponential functions Graphs and translations

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MTH108 Business Math I

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  1. MTH108 Business Math I Lecture 17

  2. Chapter 7 Exponential and Logarithmic Functions

  3. Review • Properties of exponents and radicals • Exponential functions • Special classes of exponential functions • Cases of exponential functions • Graphs and translations • Characteristics of exponential functions • Conversion to base e

  4. Today’s Topics • Logarithms • Properties • Solving exponential equations • Solving logarithmic equations • Graphs and characteristics of logarithmic functions • Some applications

  5. Logarithms Definition:A logarithm is the power x to which a base b must be raised in order to yield a given number y, i.e. Logarithm is an exponent. Consider the equation Here, 3 is the exponent which can be considered as the logarithm to the base 2 of the number 8, i.e.

  6. In general, We will restrict to the base b of positive values other than 1. Thus, logarithmic functions are related to exponential functions.

  7. Since a logarithmic function reverses the action of the corresponding exponential function, thus logarithm function is the inverse of exponential function and vice versa. Examples Exponential eq. Logarithmic eq.

  8. Examples Logarithmic eq. Exponential eq.

  9. Common bases and Notations The two most commonly used bases for logarithms are base 10 and base e, called as common logarithms and natural logarithms, respectively. Common logarithms are denoted by or more generally as, Natural logarithms are denoted as or more generally as, A logarithm having base b other than e and 10 is denoted by

  10. Properties Use: In case of computations of very large or very small numbers the use of logarithms increases efficiency. Property 1 Property 2

  11. Property 3 Property 4 Property 5

  12. Property 6 Property 7

  13. Property 8 Property 9

  14. Solving Logarithmic and Exponential Equations So far, we have studied to solve the polynomials particularly, linear, quadratic and cubic equations. Now, we will study to find the solutions of logarithmic and exponential equations. 1)

  15. 2)

  16. 3)

  17. Common mistakes

  18. Graphs of Logarithmic Functions When an unknown variable is expressed in terms of the logarithm of another variable, such a function is known as logarithmic function. A logarithmic function with base b has the form e.g.

  19. Consider the function To find the graph of , we put the values of x and calculate the corresponding values of logarithm as an output y, which is not easily determined. For this we convert the logarithmic function into the corresponding function, i.e. Now, we choose the values of y and find the corresponding values of x. Recall that the domain of an exponential function is the set of all reals and the range is the set of all positive real.

  20. Consider the function

  21. Consider the function This function can be graphed using two procedures. If values of x are given, the value of y can be evaluated by using the table or calculator, and the function can be graphed directly.

  22. Alternatively, we can graph the function by converting it into exponential form. Consider the function

  23. Summarizing the two cases, we can say that the graph of a logarithm function has one of the two shapes.

  24. Consider the function

  25. Some Applications 1) A newly created welfare agency is attempting to determine the number of analysts to hire to process welfare applications. Efficiency experts estimate that the average cost C of processing an application is a function of the number of analysts x. Specifically, the cost function is Given this logarithmic function: • Determine the average cost per application if 20 analysts are used, if 50 analysts are used. • Sketch the function.

  26. Soln.:

  27. 2) An experiment was conducted to determine the effects of elapsed time on a person’s memory. Subjects were asked to look at a picture which contained many objects. At different time intervals following this, they would be asked to recall as many objects as they could. Based on experiment, the following function was developed, where R denotes the average percentage recall and t denotes the time since studying the picture (in hours) .

  28. What is the average percentage recall 1 hour after studying the picture, after 10 hours? • Sketch the function

  29. Summary • Logarithms • Equivalent statements • Notations • Properties of Logarithms • Solving logarithmic equations • Graphs of logarithms • Some applications

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