1. A twist on Exponential Functions
Logarithms
A twist on Exponential Functions

2. First off…
Logarithm Functions are very similar to Exponential Functions
Exponential Functions – flipped the location of the variable and the real number itself
Ex. 𝑥 2 𝑣𝑒𝑟𝑠𝑢𝑠 2 𝑥
Logarithm Functions JUST FLIP THE X AND Y LOCATIONS!!!
“Flipping the variables!!!”
Ex. 𝑦= 2 𝑥 𝑣𝑒𝑟𝑠𝑢𝑠 𝑥= 2 𝑦

3. Also…what is e in terms of math????
Euler’s Number – Euler is a mathematician who came up with a “constant” number that was seen a lot in calculations for compound interest…aka $$$$
Instead of just having the number …. as a calculation, we decided that we will just use a letter in place of so many digits
e is irrational
e is the base of the “natural” logarithm
e can be calculated by using series of calculations

4. Exponential vs. Logarithm
Logarithm: The inverse of Exponential Functions
…so Logarithm = inverse of 𝑦= 𝑎 𝑥 which is: 𝑥= 𝑎 𝑦
BUT…HOW DO WE SOLVE FOR Y NOW????
Not like exponential functions because y is NOT independent…
This is why we need to know logarithms!!!!

5. Definition of Logarithm…oohhh boy!

6. When evaluating…

7. Two forms: Exponential vs. Logarithmic
Exponential Form: 𝑥= 𝑏 𝑦
Logarithmic Form: 𝑦= 𝑙𝑜𝑔 𝑏 𝑥
Both forms are the same, but usually we like STUFF without letters!
So let’s try and convert these!!!
Ex. 𝑥= 10 𝑦 Ex. 𝑦= 𝑙𝑜𝑔 5 2𝑥
Ex. 𝑥= 3 2𝑦 Ex. 3𝑥= 6 4𝑦

8. EXAMPLES!!!! 

9. EXAMPLES!!! (Cont’d) 

10. Common Log & The Properties – COMP BOOK!!!

11. Graphing Logarithmic Functions
Not hard…especially if we know how to use the inverse!!! Aka Exponential Functions
Exponential Functions Logarithmic Functions
With them being inverses, they are just “reflections” across the line
y = x
So as long as we graph exponential, then y = x…we don’t even need to come up with extra points!!!

12. Example of Graphing

13. X-Y Table…as always!!! 

15. Nature of graphing Logs…no not wood…

16. The Basics…we like.

17. Natural Logarithms… So we said that e was the natural number…
…and logarithms were in the form 𝑙𝑜𝑔 𝑎 𝑥…
…so just combine them!!!

18. Natural Logarithmic Function

19. Definition says what???
So looking at the definition, we see that the natural logarithmic functions is actually the inverse of the natural exponential function
Where did we “see” that before??? (Flip to slide 4…)

20. Graphing Natural logs…no not natural wood…

21. The properties are the same…NIIIIIICCCEE.

22. Examples of Natural Logs

23. Results of Examples

24. Main forms to take away…
When:
b y = x
Then the base b logarithm of a number x:
logb x = y
When:
y = logb x
The anti logarithm (or inverse logarithm) is calculated by raising the base b to the logarithm y:
x = logb-1(y) = b y

25. Mixed Review: 𝑥= 𝑏 𝑦 or 𝑦= 𝑙𝑜𝑔 𝑏 𝑥