The document introduces logarithms, defining them as the exponent that a fixed number (the base) must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and covers the key properties of logarithms including product, quotient, power, expanding, condensing, and the change of base formula.

1. Introduction To Logarithms

2. Our first question then must be: What is a logarithm ?

3. Definition of Logarithm Suppose b>0 and b≠1, there is a number ‘p’ such that:

4. You must be able to convert an exponential equation into logarithmic form and vice versa. So let’s get a lot of practice with this !

5. Example 1: Solution: We read this as: ”the log base 2 of 8 is equal to 3”.

6. Example 1a: Solution: Read as: “the log base 4 of 16 is equal to 2”.

7. Example 1b: Solution:

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

9. Solution:

10. Solution:

11. Solution:

12. It is also very important to be able to start with a logarithmic expression and change this into exponential form. This is simply the reverse of what we just did.

13. Example 1: Solution:

14. Example 2: Solution:

15. Okay, now you try these next three.

16. Solution:

17. Solution:

18. When working with logarithms, if ever you get “stuck”, try rewriting the problem in exponential form. Conversely, when working with exponential expressions, if ever you get “stuck”, try rewriting the problem in logarithmic form.

19. Solution: Let’s rewrite the problem in exponential form. We’re finished ! Example 1

20. Solution: Rewrite the problem in exponential form. Example 2

21. Example 3 Try setting this up like this: Solution: Now rewrite in exponential form.

22. Properties of logarithms

23. Let b, u, and v be positive numbers such that b ≠1. Product property: log b uv = log b u + log b v Quotient property: log b u/v = log b u – log b v Power property: log b u n = n log b u

24. Expanding Logarithms You can use the properties to expand logarithms. log 2 7x 3 / y= log 2 7x 3 - log 2 y = log 2 7 + log 2 x 3 – log 2 y = log 2 7 + 3 · log 2 x – log 2 y

25. Expand: log 5mn = log 5 + log m + log n Expand: log 5 8x 3 = log 5 8 + 3 · log 5 x

26. Condensing Logarithms log 6 + 2 log2 – log 3 = log 6 + log 2 2 – log 3 = log (6 ·2 2 ) – log 3 = log 24 – log 3= log 24/3= log 8

27. Condense: log 5 7 + 3 · log 5 t = log 5 7t 3 Condense: 3log 2 x – (log 2 4 + log 2 y)= log 2 x 3 /4y

28. Change of base formula: u, b, and c are positive numbers with b ≠1 and c≠1. Then: log c u = log u / log c (base 10)

29. Examples: Use the change of base to evaluate: log 3 7 = log 7 ≈ 1.771 log 3