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4.4 Evaluate Logarithms and Graph Logarithmic Functions

4.4 Evaluate Logarithms and Graph Logarithmic Functions. Part 2. Definition. Logarithms are the "opposite" of exponentials, Logs "undo" exponentials . Logs are the  inverses of exponentials. Writing Logarithms. _____________________________________________.

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4.4 Evaluate Logarithms and Graph Logarithmic Functions

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  1. 4.4 Evaluate Logarithms and Graph Logarithmic Functions Part 2

  2. Definition • Logarithms are the "opposite" of exponentials, • Logs "undo" exponentials. • Logs are the inverses of exponentials.

  3. Writing Logarithms _____________________________________________ • - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - • ____________________________________________ You read it: Log base “b” of “a” equals “c” ‘log’ is the operation b is the base a is the object of the log c is what you get when you evaluate the log

  4. Exponential Form x y = b Logarithmic Form log x y = b

  5. Evaluating logarithms now you try some! • Log 4 16 = • Log 5 1 = • Log 16 4 = • Log 3 (-1) = (Think of the graph of y = 3x) 2 0 ½ (because 161/2 = 4) undefined

  6. You should learn the following general forms!!! • Log a 1 = 0 because a0 = 1 • Log a a = 1 because a1 = a • Log a ax = x because ax = ax

  7. Common logarithms • log x = log 10 x • Understood base 10 if nothing is there.

  8. Common Logs and Natural Logs with a calculator log10 button lne button

  9. Finding Inverses • Find the inverse of: • y = log3x • By definition of logarithm, the inverse is y=3x • OR write it in exponential form and switch the x & y! 3y = x 3x = y

  10. Example 1: • Write 53 = 125 in logarithmic form. • Write log381 = 4 in exponential form.

  11. Try This: Complete the table. #1 #2 #3 #4

  12. Lets look at their graphs y = x

  13. To Evaluate Logs without a Calculator • Change the log to an exponential. 1. log2 32 = x 2. log4 2 = x

  14. Solve for x. Change the log to an exponential. 1. log2 64 = x 2. logx 343 = 3

  15. Evaluate without a calculator: Change the log to an exponential. • log 2 8 = x • log 2 1 = x 3. Find the value of k : k = log 9 3 4. Find the value of k : ½ = log k 9 5. Find the value of k : 3 = log 7 k

  16. Common Logarithms 10 • Logarithms with base ______ are called common logarithms. • Sometimes the base is assumed and not written. • Thus, if you see a log written without a base, you assume the base is _______. • The log button the calculator uses base _____. 10 10

  17. Use your calculator to evaluate: 1.71 • log 51 • log 4 • log 0.215 Which means 0.6 – 0.67

  18. Do You Know What X is? Change the exponential to a log. Then use calculator. 4. Solve for x: 10x = 728 5. Solve for x:

  19. Remember e ? The Natural Base Used for applications with CONTINUOUS growth or decay!

  20. Natural Logarithm • A natural logarithm is a logarithm with base e, denoted by ln. • A natural logarithm is the inverse of an exponential function with base e. Exponential Form Logarithmic Form

  21. Lets look at their graphs y = x

  22. Evaluate f(x)=ln x to the nearest thousandth for each value of x below: ? (see graph) 0.693 – 0.693

  23. 13. Find the inverse of y = ln(x+1) 14. Find the inverse of y = 5x . y = ex - 1 y = log5x

  24. Homework Book Pg. 147 16 - 24 all Pg. 148 13 – 21 all

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