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Logarithmic Functions

Learn how to convert between exponential and logarithmic expressions, evaluate common and natural logarithms, and apply the rules of logarithms to rewrite expressions. Discover the unique relationship between logarithms and their exponential counterparts. Explore the characteristics and transformations of logarithmic functions. Use the change-of-base formula to evaluate expressions with uncommon bases.

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Logarithmic Functions

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  1. Logarithmic Functions • TXVA: Not just a school, but an opportunity where every moment matters.

  2. Objectives • Convert between exponential and logarithmic expressions. • Evaluate common and natural logarithms. • Apply the rules of logarithms to rewrite expressions. • Evaluate expressions that have uncommon bases using the change-of-base formula.

  3. A logarithm is an Exponential function. • All logarithms have a unique relationship with their exponential counterparts – they are actually the inverses of their exponential counterparts. • can be written as • I use this to help me remember, “base to the answer power = the guts”. • Try:

  4. Use the rules of logarithms to expand: . Use the rule of logarithms to expand:

  5. Common Log & Natural Log • A logarithmic function with base 10 is called • a Common Log. • Denoted: • A logarithmic function with base e is called • a Natural Log. • Denoted: *Note there is no base written.

  6. Find y if Explain your thinking.

  7. Find y if Explain your thinking. If , what is

  8. Write the expression as a single logarithm. Write the expression as a single logarithm.

  9. Solve for x if Solve for x if

  10. True or False?

  11. Write in exponential form: Solve:

  12. Rewrite the following expression using a single logarithm.

  13. Solve: If , what is the value of Let , what is the value of

  14. Rewrite the following expression as a single logarithm: Complete the sentence is the power to which ___________must be raised to produce a value of 81.

  15. If and what can you say about the relationship between A and C?

  16. If , what is Solve for :

  17. Solve for :

  18. The inverse function of an exponential function is the logarithmic function. • For all positive real numbers x and b, b>0 and b1, y=logbx if and only if x=by. • The domain of the logarithmic function is • The range of the function is • Since the log function is the inverse of the • exponential function, their graphs are symmetric • with respect to the line y=x.

  19. **Remember that since the log function is the inverse of the exponential function, we can simply swap the x and y values of our important points!

  20. Logarithmic Functions where b>1 are ___________, one-to-one functions. • Logarithmic Functions where 0<b<1 are __________, one-to-one functions. • The parent form of the graph has an x-intercept at (1,0) and passes through _____and _______ • There is a vertical asymptote at _______. • The value of b determines the flatness of the curve. • The function is neither even nor odd. There is ______ symmetry. • There is _____ local extrema.

  21. More Characteristics of • The domain is • The range is • End Behavior: • As • As • The x-intercept is • The vertical asymptote is • There is no y-intercept. • There are no horizontal asymptotes. • This is a ___________, __________ function. • It is concave _________.

  22. Graph: Domain: Range: x-intercept: Important Points: Vertical Asymptote: Inc/dec? Concavity? End Behavior:

  23. Graph: *Reflects @ x-axis. Important Points: Domain: Range: x-intercept: Vertical Asymptote: Inc/dec? Concavity?

  24. Transformations Horizontal Reflect * Reflect Vertical Vertical. Vertical Vertical Domain: Range: x-intercept: Domain: Range: x-intercept: Domain: Range: x-intercept: Vertical Asymptote: Vertical Asymptote: Vertical Asymptote: Inc/dec? Inc/dec? Inc/dec? Concavity? Concavity? Concavity?

  25. The asymptoteof a logarithmic function of this form is the line To find an x-intercept in this form, let y=o in the equation To find a y-intercept in this form, let x=o in the equation Since this is not possible, there is No y-intercept. is the vertical asymptote. is the x-intercept

  26. Check it out!  is the vertical asymptote. Horizontal Horizontal  is the x-intercept. Since this is not possible, there is No y-intercept.  Domain: Range: x-intercept: Vertical Asymptote: Inc/dec? Concavity?

  27. Transformations Common Log to find the V.A. Let Horizontal. Horizontal is the V.A. Let to find the x-intercept. is the x-intercept. Domain: Range: Inc/Dec: Concavity: Let to find the y-intercept. is the y-intercept.

  28. Transformations Natural Log Let to find the V.A. Reflect. Vertical. Horizontal Horizontal. Vertical is the V.A. Let to find the x-intercept. is the x-intercept. Let to find the y-intercept. Domain: Range: x-intercept: Inc/dec? Concavity? is the y-intercept.

  29. Change of Base Formula Use this formula for entering logs with bases other than 10 or e in your graphing calculator. So, if you wanted to graph , you would enter in your calculator. Either the natural or common log may be used in the change of base formula. So, you could also enter in Your calculator.

  30. Evaluate Evaluate

  31. If what is x?

  32. Tell me what you know….

  33. Tell me what you know…. Domain: Range: Domain: Range:

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