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Chapter 5

Chapter 5. Logarithmic, Exponential, and Other Transcendental Functions. A logarithm is an exponent!. For x  0 and 0  a  1, y = log a x if and only if x = a y . The function given by f ( x ) = log a x i s called the logarithmic function with base a.

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Chapter 5

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  1. Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

  2. A logarithm is an exponent! For x 0 and 0  a  1, y = loga x if and only if x = ay. The function given by f(x) = loga x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y= loga x is equivalent to x =ay A logarithmic function is the inverse function of an exponential function. Exponential function: y = ax Logarithmic function: y = logax is equivalent to x = ay

  3. y (x  0, e 2.718281) x 5 –5 y = ln x is equivalent to ey = x y = ln x The function defined by f(x) = logex = ln x is called the natural logarithm function. In Calculus, we work almost exclusively with natural logarithms!

  4. Definition of the Natural Logarithmic Function

  5. Theorem 5.1 Properties of the Natural Logarithmic Function

  6. Natural Log

  7. Natural Log Passes through (1,0) and (e,1). You can’t take the log of zero or a negative. (Same graph 1 unit right)

  8. Theorem 5.2 Logarithmic Properties

  9. Properties of Natural Log: Expand: Write as a single log:

  10. Properties of Natural Log: Expand: Write as a single log:

  11. Definition of e

  12. Theorem 5.3 Derivative of the Natural Logarithmic Function

  13. Derivative of Logarithmic Functions The derivative is Notice that the derivative of expressions such as ln|f(x)| has no logarithm in the answer. Example: Solution:

  14. Example

  15. Example

  16. Example Product Rule

  17. Example

  18. Example

  19. Example

  20. Example

  21. Theorem:

  22. Theorem:

  23. Theorem 5.4 Derivative Involving Absolute Value

  24. Try Logarithmic Differentiation.

  25. 4. Show that is a solution to the statement .

  26. 4. Show that is a solution to the statement .

  27. Find the equation of the line tangent to: at (1, 3) At (1, 3) the slope of the tangent is 2

  28. Find the equation of the tangent line to the graph of the function at the point (1, 6).

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