This document discusses properties of logarithms, including: 1) Logarithms with the same base "undo" each other according to the inverse function relationship between logarithms and exponents. 2) Logarithmic expressions can be expanded using properties to write them as sums or differences of individual logarithmic terms, or condensed into a single logarithm. 3) The change of base formula allows converting between logarithms with different bases, with common uses being to change to base 10 or the base of natural logarithms.

1. Properties of Logarithms

2. Since logs and exponentials of the same base are inverse functions of each other they “undo” each other. Remember that: This means that: inverses “undo” each each other = 5 = 7

3. CONDENSED EXPANDED Properties of Logarithms = = = = (these properties are based on rules of exponents since logs = exponents) 3. 2. 1.

4. Using the log properties, write the expression as a sum and/or difference of logs (expand). using the second property: When working with logs, re-write any radicals as rational exponents. using the first property: using the third property:

5. Using the log properties, write the expression as a single logarithm (condense). using the third property: using the second property: this direction this direction

11. Write the following expression as a single logarithm.

13. More Properties of Logarithms This one says if you have an equation, you can take the log of both sides and the equality still holds. This one says if you have an equation and each side has a log of the same base , you know the "stuff" you are taking the logs of are equal.

14. (2 to the what is 8?) (2 to the what is 16?) (2 to the what is 10?) There is an answer to this and it must be more than 3 but less than 4, but we can't do this one in our head. Let's put it equal to x and we'll solve for x . Change to exponential form. use log property & take log of both sides (we'll use common log) use 3rd log property solve for x by dividing by log 2 use calculator to approximate Check by putting 2 3.32 in your calculator (we rounded so it won't be exact)

15. Common and Natural Logarithms

20. Use the Change-of-Base Formula and a calculator to approximate the logarithm. Round your answer to three decimal places. Since 3 2 = 9 and 3 3 = 27, our answer of what exponent to put on 3 to get it to equal 16 will be something between 2 and 3. put in calculator

26. SOLVE

27. SOLVE

28. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

29. Change-of-Base Formula The base you change to can be any base so generally we’ll want to change to a base so we can use our calculator. That would be either base 10 or base e . “common” log base 10 “natural” log base e Example for TI-83 If we generalize the process we just did we come up with the: LOG LN