1. Introduction To Logarithms

2. Introduction to Logarithmic Functions You were introduced to inverse functions. Inverse functions is the set of ordered pair obtained by interchanging the x and y values. f(x) f -1 (x) GRAPHS OF EXPONENTIALS AND ITS INVERSE

3. Introduction to Logarithmic Functions Inverse functions can be created graphically by a reflection on the y = x axis. y = x f(x) f -1 (x) GRAPHS OF EXPONENTIALS AND ITS INVERSE

4. Introduction to Logarithmic Functions A logarithmic function is the inverse of an exponential function Exponential functions have the following characteristics: Domain: {x є R} Range: {y > 0}

5. Introduction to Logarithmic Functions Let us graph the exponential function y = 2 x Table of values: GRAPHS OF EXPONENTIALS AND ITS INVERSE

6. Introduction to Logarithmic Functions Let us find the inverse the exponential function y = 2 x Table of values: GRAPHS OF EXPONENTIALS AND ITS INVERSE

7. Introduction to Logarithmic Functions When we add the function f(x) = 2 x to this graph, it is evident that the inverse is a reflection on the y = x axis f -1 (x) f(x) GRAPHS OF EXPONENTIALS AND ITS INVERSE f(x)f -1 (x)

8. Introduction to Logarithmic Functions You will find the inverse of an exponential algebraically Write the process in your notes FINDING THE INVERSE OF AN EXPONENTIAL y = a x Interchange x  y x = a y We write these functions as: x = a y y = log a x x = a y base exponent a x y = log a x base exponent

9. Introduction to Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL y = a x Exponential Function Inverse of the Exponential Function x y log Logarithmic Form

10. Introduction to Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 4 3 = 64 b) 4 5 = 256 c) 2 7 = 128 d) (1/3) x =27

11. Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log 2 64=6 b) log 25 5=1/2 c) log 8 1=0 d) log 1/3 9=2 ANSWERS

12. Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log 2 64=6 2 6 = 64 25 1/2 = 5 8 0 = 1 (1/3) 2 = 1/9 b) log 25 5=1/2 c) log 8 1=0 d) log 1/3 9=2

13. Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log 1/3 27 = x b) log 5 x = 3 c) log x (1/9) = 2 d) log 3 x = 0 ANSWERS

14. Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log 1/3 27 = x b) log 5 x = 3 c) log x (1/9) = 2 d) log 3 x = 0 (1/3) x = 27 (1/3) x = (1/3) -3 x = -3 5 3 = x x = 125 x 2 = (1/9) x = 1/3 3 0 = x x = 1

15. Introduction to Logarithmic Functions BASE 10 LOGS Scientific calculators can perform logarithmic operations. Your calculator has a LOG button. This button represents logarithms in BASE 10 or log 10 Example 4) Use your calculator to find the value of each of the following: a) log 10 1000 b) log 50 c) log -1000 =3=3 = 1.699 = Out of Domain

16. What is a logarithm ?

17. Solution: We read this as: ”the log base 2 of 8 is equal to 3”.

18. Solution: Read as: “the log base 4 of 16 is equal to 2”.

19. Solution:

20. Okay, so now it’s time for you to try some on your own.

21. Solution:

26. Okay, now you try these next three.

27. Solution:

30. Our final concern then is to determine why logarithms like the one below are undefined. Can anyone give us an explanation ?

31. One easy explanation is to simply rewrite this logarithm in exponential form. We’ll then see why a negative value is not permitted. First, we write the problem with a variable. Now take it out of the logarithmic form and write it in exponential form. What power of 2 would gives us -8 ? Hence expressions of this type are undefined.