1. Warm Up – 4.28 - Monday 1. Write each logarithm as an exponential:
log =𝑥−1 B) ln 𝑦=5𝑥
2. Write each exponential as a logarithm.
4 𝑥 =15 B) 𝑒 𝑥−3 =8
3. Solve the logarithmic equation.
log 5 𝑥+1 = log
log 2 3𝑦+1 = log 2 (2𝑦)

2. Rational Exponents
When I raise an exponent to a fraction, that is the same as taking the root of the denominator.
Example: 4 5/3 =
Example: 7 −4/5 =

3. Rational Exponents Classwork

4. Rules of Logarithms log 𝑎 𝑏 + log 𝑎 𝑐 = log 𝑎 (𝑏∙𝑐)
THESE RULES ONLY APPLY IF THE BASE IS THE SAME!!!
log 𝑎 𝑏 + log 𝑎 𝑐 = log 𝑎 (𝑏∙𝑐)
log 𝑎 𝑏 − log 𝑎 𝑐 = log 𝑎 𝑏 𝑐
log 𝑎 𝑏 𝑐 =𝑐∙ log 𝑎 𝑏

5. Expanding Logarithms
These problems are finished when each logarithm has only one number or integer on the inside.
Example:
log 2 𝑥 + log 2 ⁡(𝑦∙𝑧)
The first log is as simplified as it gets. The second log has two variables and needs to be expanded.

6. Example #1
log 2 𝑥 + log 2 ⁡(𝑦∙𝑧)

7. Example #1 - Solution log 2 𝑥 + log 2 ⁡(𝑦∙𝑧)
When we multiply on the inside we add the logs.
log 2 𝑥 +( log 2 𝑦 + log 2 (𝑧))
Notice the parentheses. These are not necessary here because there is a positive in front of them.
log 2 𝑥 + log 2 𝑦 + log 2 𝑧

8. Example #2
log 2 𝑥𝑦 𝑧

9. Example #2 - Solution log 2 𝑥𝑦 𝑧
Take care of the division before the multiplication.
log 2 𝑥𝑦 − log 2 𝑧
(log 2 𝑥 + log 2 𝑦 )− log 2 𝑧
We can drop the parenthesis because there is no negative in front.
log 2 𝑥 + log 2 𝑦 − log 2 𝑧

10. Example #3
log 4 𝑥 𝑦𝑧

11. Example #3 - Solution log 4 𝑥 𝑦𝑧 log 4 𝑥 − log 4 𝑦𝑧
There is a minus sign in front of the parenthesis so we change the signs on the inside.
log 4 𝑥 − log 4 𝑦 − log 4 (𝑧)

12. Example #4
log

13. Example #4 - Solution log 3 2 9 3
When everything is raised to an exponent, deal with the exponent first.
3 log
3( log − log 3 (9))
Distribute the 3.
3 log −3 log 3 9

14. Example #5
log 5 (𝑥∙ 𝑦 7 )

15. Example #5 - Solution log 5 (𝑥∙ 𝑦 7 )
When only one base is raised to an exponent, deal with the multiplication first!
log 5 𝑥 + log 5 ( 𝑦 7 )
log 5 𝑥 +7 log 5 𝑦