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Logarithms

Logarithms. Section 3.3a!!!. First, remind me…. What does the horizontal line test tell us???. More specifically, what does it tell us about the function. This function has an inverse that is also a function!!!. This inverse is called the logarithmic function with base b. Notation:.

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Logarithms

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  1. Logarithms Section 3.3a!!!

  2. First, remind me… What does the horizontal line testtell us??? More specifically, what does it tell us about the function This function has an inverse that is also a function!!! This inverse is called the logarithmic function with base b. Notation:

  3. Changing Between Logarithmic and Exponential Form If x > 0 and 0 < b = 1, then if and only if Important Note: The “linking statement” says that a logarithm is an exponent!!!

  4. Basic Properties of Logarithms For 0 < b = 1, x > 0, and any real number y, 1. because 2. because 3. because 4. because

  5. Evaluating Logarithms Evaluate each of the following. 1. 4. 2. 5. 3. 6.

  6. What’s true about the (x, y) pairs and graphs of inverse functions? –3 1/8 1/8 –3 –2 1/4 1/4 –2 –1 1/2 1/2 –1 0 1 1 0 1 2 2 1 2 4 4 2 3 8 8 3 Now, let’s plot these points and discuss the graphs…

  7. Common Logarithms Common Logarithm – logarithm with a base of 10 (very commonly used because of our base 10 number system!) For common logarithms, we can drop the subscript: if and only if

  8. Basic Properties of Common Logarithms Let x and y be real numbers with x > 0. 1. because 2. because 3. because 4. because

  9. More Evaluating Logarithms Evaluate each of the following. 1. 2. 3. 4. Note: The LOG key on your calculator refers to the common logarithm…

  10. Using Your Calculator Use a calculator to evaluate the logarithmic expression if it is defined, and check your result by evaluating the corresponding exponential expression. b/c 1. b/c 2. is undefined  can you explain why ? 3.

  11. Solving Simple Logarithmic Equations Solve the given equations by changing to exponential form. 1. 2. Exp. Form: Exp. Form:

  12. What is the definition of the natural base???

  13. Natural Logarithms Natural Logarithm – a logarithm with base e Notation: ln That is, Back to our inverse relationship: if and only if

  14. Basic Properties of Natural Logarithms Let x and y be real numbers with x > 0. 1. because 2. because 3. because 4. because

  15. Cool Practice Problems Evaluate each of the following without a calculator. 1. 3. 2. Note: The LN key on your calculator refers to the natural logarithm…

  16. Cool Practice Problems Use a calculator to evaluate the given logarithmic expressions, if they are defined, and check your result by evaluating the corresponding exponential expression. 1. because 2. is undefined!!! Why??? 3. because

  17. Cool Practice Problems Solve each of the given equations by changing them to exponential form. 2. 1. 1 x = 100,000 x = = 0.368… e 3. 4. 1 2.5 x = e = 12.182… x = = 0.01 100

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