1. The Exponential and Logarithmic Functions
Tada!

2. 3.1a The Exponential Model The Parent Function: 𝑓 𝑥 = 𝐵 𝑥
The Transformations:
𝑓 𝑥 = 𝑎𝐵 𝑏(𝑥−ℎ) +𝑘
The VIP’s:
Other stuff:
𝑦=0
(0,1)
(1,𝐵)

4. 3.1b The Exponential Model Why does it look like it does? 𝑓 𝑥 = 1 3 𝑥
𝑓 𝑥 = 𝑥
𝑓 𝑥 = 𝑥
𝑓 𝑥 = 2 𝑥 x y -2 -1 1 2 3 x y -2 -1 1 2 3 x y -2 -1 1 2 3

5. 3.1b The Exponential Model Why does it look like it does?-2
128 -1 32 8 1 2
1 2 3
1 8 x y -2
4 9 -1
4 3 4 1 12 2 36 3
108
𝑓 𝑥 = 𝑥
𝑓 𝑥 = 4 3 𝑥

6. 3.1b Base as an obvious multiplier 𝑓(𝑥)=𝑎 (𝑏𝑎𝑠𝑒) 𝑥 𝑎=𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
(doubled, cut in half, etc.)
𝑓(𝑥)=𝑎 (𝑏𝑎𝑠𝑒) 𝑥
𝑎=𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
(𝑦−𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑤ℎ𝑒𝑛 𝑛𝑜 ℎ 𝑜𝑟 𝑘 𝑚𝑜𝑣𝑒𝑚𝑒𝑛𝑡)
𝑏𝑎𝑠𝑒=𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟
(𝑏𝑎𝑠𝑒)>1 → 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑔𝑟𝑜𝑤𝑡ℎ
Assuming a is greater than 0
0<(𝑏𝑎𝑠𝑒)<1 → 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑒𝑐𝑎𝑦
Assuming a is greater than 0

7. 3.1b The Natural Base 𝑒=2.71828182845904523 𝑒= lim 𝑥→∞ 1+ 1 𝑥 𝑥
Leonhard Euler
Andrew Jackson 𝑒=
𝑒= lim 𝑥→∞ 𝑥 𝑥
“e…is…a little less than 3”
𝑤 𝑏 𝑣 = 𝑏 𝑣 𝑤
The Logistic Function
𝑎 −𝑛 = 1 𝑎 𝑛
𝑓(𝑥)= 1 1+ 𝑒 −𝑥

8. 3.2a Base as an obvious multiplier Base as a percentage rate
(Population, radioactive decay)
3.2a
Base as an obvious multiplier
(doubled, cut in half, etc.)
Base as a percentage rate
𝑓(𝑥)=𝑎 (𝑏𝑎𝑠𝑒) 𝑥
𝑃(𝑡)= 𝑃 0 (1+𝑟) 𝑡
𝑎=𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
𝑃 0 =𝑎=𝑖𝑛𝑖𝑡𝑎𝑙 𝑣𝑎𝑙𝑢𝑒
𝑏𝑎𝑠𝑒=𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟
𝑟=𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑟𝑎𝑡𝑒
Ex’s 1-4 in book pg. 265
𝑡=𝑡𝑖𝑚𝑒, 𝑢𝑠𝑢𝑎𝑙𝑙𝑦 𝑦𝑒𝑎𝑟𝑠
𝑏𝑢𝑡 𝑝𝑎𝑦 𝑎𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛
(𝑏𝑎𝑠𝑒)>1 → 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑔𝑟𝑜𝑤𝑡ℎ ←(1+𝑟)>1
0<(𝑏𝑎𝑠𝑒)<1 → 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑒𝑐𝑎𝑦 ←0<(1−𝑟)<1

9. 3.3a The Logarithmic Model What is it? A logarithm IS the exponent!
𝐥𝐨𝐠 𝒃 𝒙=𝒚 is equivalent to 𝒙=𝒃 𝒚
log 2 8 =
log =
Some Properties
log =
log 𝑏 1=0
log 𝑏 𝑏=1
log 𝑏 𝑏 𝑦 =𝑦
𝑏 log 𝑏 𝑥 =𝑥
log =
𝑤 𝑏 𝑣 = 𝑏 𝑣 𝑤
log 4 1 =
𝑎 −𝑛 = 1 𝑎 𝑛
log 7 7 =

10. 3.3a The Common Log log 10 𝑥 = log 𝑥 The Natural Log log 𝑒 𝑥 = ln 𝑥
Base is 10
The Natural Log
log 𝑒 𝑥 = ln 𝑥
“e…is…a little less than 3”
Base is e

11. 3.3b The Logarithmic Model The Parent Function: 𝑓 𝑥 = log 𝐵 𝑥
The Transformations:
𝑓 𝑥 = alog 𝐵 (𝑏 𝑥−ℎ ) +𝑘
The VIP’s:
Other stuff:
𝑥=0
(1,0)
(𝐵,1)

14. Properties of Logarithms
log 𝑏 𝑚𝑛 = log 𝑏 𝑚 + log 𝑏 𝑛
log 𝑏 𝑚 𝑛 = log 𝑏 𝑚 − log 𝑏 𝑛
log 𝑏 𝑚 𝑐 =𝑐 log 𝑏 𝑚
𝑤 𝑏 𝑣 = 𝑏 𝑣 𝑤
log 𝑏 1=0
log 𝑏 𝑏=1
log 𝑏 𝑏 𝑦 =𝑦
𝑏 log 𝑏 𝑥 =𝑥
𝑎 −𝑛 = 1 𝑎 𝑛

15. Solving Logarithms or for variables in the exponent
Make the bases the same (sometimes works)
𝒃 𝒚 = 𝒃 𝒚 𝐥𝐨𝐠 𝒃 𝒙 = 𝐥𝐨𝐠 𝒃 𝒙
Rewrite (works nearly all of the time) 𝐥𝐨𝐠 𝒃 𝒙=𝒚 is equivalent to 𝒙=𝒃 𝒚
Take the log of both sides (always works)

16. Graphing Logarithms/Exponents
Exponential
Logarithmic
𝑦=𝑎 log 𝑏 (𝑥−ℎ) +𝑘
𝑓(𝑥)=𝑎 𝑏 𝑥−ℎ +𝑘
𝑓(𝑥)=𝑎 𝑏 𝑥
𝑃(𝑡)=𝑎 (1+𝑟) 𝑡
𝑎=𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑎𝑙𝑢𝑒 𝑟=% 𝑏=𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑏> 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑔𝑟𝑜𝑤𝑡ℎ →𝑏=(1+𝑟) 0<𝑏< 𝑒𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑒𝑐𝑎𝑦 →𝑏=(1−𝑟)
Simplifying and Solving
Make the bases the same (sometimes works)
𝒃 𝒚 = 𝒃 𝒚 𝐥𝐨𝐠 𝒃 𝒙 = 𝐥𝐨𝐠 𝒃 𝒙
Rewrite (always works) 𝐥𝐨𝐠 𝒃 𝒙=𝒚 is equivalent to 𝒙=𝒃 𝒚