1. Properties of Logarithms

2. Rules of Logarithms If M and N are positive real numbers and b is ≠ 1:
The Product Rule:
logbMN = logbM + logbN
(The logarithm of a product is the sum of the logarithms)
Example: log4(7 • 9) = log47 + log49
Example: log (10x) = log10 + log x

3. Rules of Logarithms If M and N are positive real numbers and b ≠ 1:
The Product Rule:
logbMN = logbM + logbN
(The logarithm of a product is the sum of the logarithms)
Example: log4(7 • 9) = log47 + log49
Example: log (10x) = log10 + log x
You do: log8(13 • 9) =
You do: log7(1000x) =
log813 + log89
log log7x

4. Rules of Logarithms If M and N are positive real numbers and b ≠ 1:
The Quotient Rule
(The logarithm of a quotient is the difference of the logs)
Example:

5. Rules of Logarithms If M and N are positive real numbers and b ≠ 1:
The Quotient Rule
(The logarithm of a quotient is the difference of the logs)
Example:
You do:

6. Rules of Logarithms If M and N are positive real numbers, b ≠ 1, and p is any real number:
The Power Rule:
logbMp = p logbM
(The log of a number with an exponent is the product of the exponent and the log of that number)
Example: log x2 = 2 log x
Example: ln 74 = 4 ln 7
You do: log359 =
Challenge:
9log35

7. Prerequisite to Solving Equations with Logarithms
Simplifying
Expanding
Condensing

8. Simplifying (using Properties)
log94 + log96 = log9(4 • 6) = log924
log 146 = 6log 14
You try: log log1612 =
You try: log316 + log24 =
You try: log log 3 =
log163
Impossible!
log 5

9. Using Properties to Expand Logarithmic Expressions
Use exponential notation
Use the product rule
Use the power rule

10. Expanding

11. Condensing
Condense:
Product Rule
Power Rule
Quotient Rule

12. Condensing
Condense:

13. Bases Everything we do is in Base 10.
We count by 10’s then start over. We change our numbering every 10 units.
In the past, other bases were used.
In base 5, for example, we count by 5’s and change our numbering every 5 units.
We don’t really use other bases anymore, but since logs are often written in other bases, we must change to base 10 in order to use our calculators.

14. Change of Base Examine the following problems:
log464 = x
we know that x = 3 because 43 = 64, and the base of this logarithm is 4
log 100 = x
If no base is written, it is assumed to be base 10
We know that x = 2 because 102 = 100
But because calculators are written in base 10, we must change the base to base 10 in order to use them.

15. Change of Base Formula Example log58 =
This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!