1. Logarithmic Expressions and the 5 Properties of Logarithms
2/19
Module 5 is due at 11:59 pm on Friday night.
Logarithmic Expressions and the 5 Properties of Logarithms
We will explore the properties of logarithms with the ultimate goal solving exponential and logarithmic equations.

2. Take each group of 3 numbers and create a true logarithmic equation.
1.) 81, 2, 9
2.) −𝟑, 𝟔, 𝟏 𝟐𝟏𝟔
3.) 4, 4, 1
4.) 256, 4, 4

3. Can we solve simple logarithmic equations?

4. 5 Properties of Logarithms
The following 5 Properties of Logarithms will give us the capabilities to solve exponential and logarithmic equations.

5. 1st Property of Logarithms: When the argument of a log is 1.
Anytime the argument of a log is 1, then the logarithmic equation will equal 0.
This is because anything raised to the zero power is 1.
log 𝑥 1 =0
because
𝑥 0 =1

6. 2nd Property of Logarithms: When the base and the argument of a log are the same.
When the base and the argument of a log are the same, then the logarithmic equation will equal 1.
This is because any number to the first power is “itself” or the same number.
log 4 4 =1
because
4 1 =4

7. 3rd Property of Logarithms: Product Rule of Logarithms
The Product Rule of Logarithms allows us to rewrite multiple logs with the SAME base as one log when they are separated by a plus sign.
log 3 𝑥 + log 3 𝑦 = log 3 𝑥𝑦 or
log log 6 3 = log 6 36 or
log 2 10𝑥 = log log 2 𝑥

8. 4th Property of Logarithms: Quotient Rule of Logarithms
The Quotient Rule of Logarithms allows us to rewrite multiple logs with the SAME base as one log when they are separated by a minus sign.
log 3 𝑥 − log 3 𝑦 = log 3 𝑥 𝑦 or
log − log 6 3 = log 6 4 or
log 𝑥 = log − log 2 𝑥

9. 5th Property of Logarithms: Power Rule of Logarithms
The Power Rule of Logarithms is an especially powerful tool. It allows us to move an exponent in the argument to the front of the log and make the operation multiplication.
log =2 log 5 25 or
log 𝑥 =𝑥 log 6 15 or
5 log 2 𝑥= log 2 𝑥 5