1. Exponential & Log Functions
Unit 13 Day 4

2. Warm up!
Solve: 𝑥+2 ∗ 216 3𝑥 = 1 216
A total of $12,000 is invested at an annual rate of 9%. Find the balance after 5 years if it is compounded a)Quarterly b)Monthly c)Continuously.
Describe the transformations and graph:
𝑦= 𝑥−3 −2
𝑓 𝑥 =− 𝑙𝑜𝑔 3 𝑥−2 +4
X=-1/6
A=$18, A=$18, A=$18,819.75
a=2, b=1, h=3, k=0
a=-1, b=1, h=2, k=4

3. Warm up answers
𝑦= 𝑥−3 −2 𝑓 𝑥 =− 𝑙𝑜𝑔 3 𝑥−2 +4

4. Think-Ink-Pair-Share
How would you solve…
9 𝑥 =49

5. Changing Forms
To change between exponential and logarithmic forms…

6. Changing Forms Examples
𝑙𝑜𝑔 4 2=𝑥
4 𝑥 =2
9 𝑥 =3
𝑙𝑜𝑔 9 3=𝑥
ln 𝑥 =4
𝑒 4 =𝑥

7. You Try! =8
𝑙𝑜𝑔 4 8= 3 2
𝑙𝑜𝑔 =−2
4 −2 = 1 16

8. Logarithmic Properties
loga1 = 0 (a0=1)
Inverse Property loga ax = x
loga a = 1 (a1=a)
One-to-one property loga x = loga y then x = y
also applies to exponential functions
ax = ay then x = y

9. Using the Properties Evaluate the following expressions 𝑙𝑛 𝑒 2 =
𝑙𝑛 𝑒 2 = 2
𝑙𝑜𝑔 = -1
ln 𝑒 3𝑥+4𝑦 =
3x +4y

10. You Try! Evaluate the following expressions 𝑒 𝑙𝑛𝑥 = 𝑙𝑜𝑔 5 5 3 =
𝑙𝑜𝑔 =
1.5
6 𝑙𝑜𝑔 6 20𝑡 =
20t

11. Solving Exponential & Logarithmic Equations Example 1
Solve for x
𝑙𝑜𝑔 3 𝑥=4
Change forms
3 4 =𝑥 (on non-calculator part of test, this is an acceptable answer)
𝑥=81

12. Solving Exponential & Logarithmic Equations Example 2 & 3
Solve for x
ln 𝑥 =−3
Use the inverse property
𝑥= 𝑒 −3 (exact answer/form)
x=0.050
𝑒 𝑥 =7
𝑥= ln 7 𝑜𝑟 1.95

13. Solving Exponential & Logarithmic Equations Example 4
Solve for x
ln 𝑥 − ln 3 =0
ln 𝑥 = ln (3)
𝑥=3

14. Solving Exponential & Logarithmic Equations Example 5
How would you solve?
𝑒 2𝑥 −3 𝑒 𝑥 +2=0
𝐹𝑎𝑐𝑡𝑜𝑟!
𝑒 𝑥 −1 𝑒 𝑥 −2 =0
𝑒 𝑥 −1 = 𝑒 𝑥 −2 =0
𝑒 𝑥 = 𝑒 𝑥 =2
𝑥= ln 1 =0 𝑥= ln 2 =0.693

15. You Try! Solve for x. 𝑙𝑜𝑔 2 𝑥+3 − 𝑙𝑜𝑔 2 3𝑥−4 =0
𝑙𝑜𝑔 2 𝑥+3 − 𝑙𝑜𝑔 2 3𝑥−4 =0
7/2 or 3.5
log 𝑥=3 *remember log x = log10x
x=1000
𝑙𝑜𝑔 2 𝑥 2 +2𝑥 = 𝑙𝑜𝑔 2 𝑥+6
x=-3 x=2
𝑒 2𝑥 −3 𝑒 𝑥 +2=0
x=ln4
2𝑥𝑙𝑛 𝑥 +𝑥=0
𝑥=0 𝑥=𝑒 −1

16. Solving Exponential & Logarithmic Equations Example 5Extraneous Solutions
Extraneous solutions occur when a solution is x = ln(c) and c ≤ 0
Solve for x
𝑒 2𝑥 −3 𝑒 𝑥 −4=0
𝑒 𝑥 +1 𝑒 𝑥 −4 =0
𝑒 𝑥 +1 = 𝑒 𝑥 −4 =0
𝑒 𝑥 =− 𝑒 𝑥 =4
𝑥= ln − 𝒙= 𝐥𝐧 𝟒
Extraneous!

17. Applications Example 1
If you invest $500 at 6.75% compounded continuously, how long until it doubles?
𝐴=𝑃 𝑒 𝑟𝑡
1000=500 𝑒 𝑡
2= 𝑒 (𝑡)
ln 2=0.0675𝑡
t=10.27 years

18. You Try! Applications
If you invest $1000 for 7 years and now have $ , what was your interest rate (compounded continuously)?
r = 5.99%

19. Exit Ticket
On a scale of 1 to 5 with 1 being “were you speaking English?” 5 being “I totally got this!” How would you rate your understanding of today’s lesson? Explain in a few words.