1. Lesson 9 – 5 Exponential Equations & Inequalities
Pre-calculus

2. Learning Objective To solve exponential equations
To solve continuous interest problems

3. Logarithmic Vocabulary
Consider:
log 260
= log (2.6 𝑥 )
= log log 10 2 =
=2.4150
mantissa
characteristic
Also log 0.26
= log (2.6 𝑥 10 −1 )
= log log −1
= (−1)
=−0.5850
mantissa
characteristic

4. Logarithmic Vocabulary
1. Underline the mantissa & circle the characteristic
Logarithmic Vocabulary
log 425 =2.6284
If we are given log 𝑥 or ln 𝑥 , we can find 𝑥 using our calculators.
We will use 10 𝑥 or 𝑒 𝑥
Let’s practice some simple ones…get those calculators ready!

5. Logarithmic Equations
Solve for 𝑥 to the nearest hundredth.
Logarithmic Equations
2. log 𝑥 =3.2274 =𝑥 𝑥=
Around the world!
3. 2log 𝑥 =2.6419
log 𝑥 = =𝑥
𝑥=20.94
4. ln 𝑥 −3=5.7213
ln 𝑥 =8.7213
𝑒 =𝑥 𝑥=

6. Check – up Solve for 𝑥 to the nearest hundredth. 1. log 𝑥 =2.3010
𝑥=199.99

7. Exponential Equations
To solve an exponential equation:
Exponential Equations
1. Isolate the exponential expression
2. Take the logarithm of both sides of the equation
3. Verify all answers! (by substitution in original)

8. Exponential Equations
Solve for 𝑥 to nearest hundredth.
Exponential Equations
𝑥−1 −27=0
4 2𝑥−1 =27
log 4 2𝑥−1 = log 27
(2𝑥−1) log 4 = log 27
2𝑥 log 4 − log 4 = log 27
2𝑥 log 4 = log 27 + log 4
𝑥= log 27 + log 4 2 log 4
𝑥=1.69

9. Exponential Equations
Solve for 𝑥 to nearest hundredth.
Exponential Equations
−2𝑥+1 −0.71=0
3 16 −2𝑥+1 =0.71
16 −2𝑥+1 =
Move decimal 2 places
log 16 −2𝑥+1 = log
(−2𝑥+1) log 16 = log
Distribute
−2𝑥 log 16 + log 16 = log
−2𝑥 log 16 = log − log 16
𝑥= log − log 16 −2 log 16
𝑥=0.76

10. Exponential Equations
Solve for 𝑥 to nearest hundredth.
Exponential Equations
𝑥−1 =4( 2 7𝑥+3 )
log( 𝑥−1 ) = log (4 2 7𝑥+3 )
Expand logs
log 3 + log 7 4𝑥−1 = log 4 + log ( 2 7𝑥+3 )
log 3 + 4𝑥−1 log 7 = log 4 +(7𝑥+3) log 2
log 3 +4𝑥 log 7 − log 7 = log 4 +7𝑥 log 2 +3 log 2
4𝑥 log 7 −7𝑥 log 2 = log 4 +3 log 2 − log 3 + log 7
𝑥(4 log 7 −7 log 2 )= log 4 +3 log 2 − log 3 + log 7
𝑥= log 4 +3 log 2 − log 3 + log 7 4 log 7 −7 log 2
𝑥=1.47

11. Solve for 𝑥 to nearest hundredth.
Check – up
𝑥 = 6 𝑥+1
𝑥=1.83

12. Exponential Equations
Solve for 𝑥 to nearest hundredth.
Exponential Equations
8. 𝑒 𝑥 𝑒 𝑥 −1 =5
𝑒 𝑥 =5( 𝑒 𝑥 −1)
𝑒 𝑥 =5 𝑒 𝑥 −5
−4 𝑒 𝑥 =−5
𝑒 𝑥 = 5 4
ln 𝑒 𝑥 = ln 5 4
𝑥= ln 5 4
𝑥=0.22

13. Exponential Equations
Solve for 𝑥 to nearest hundredth.
Exponential Equations
9. 2 𝑒 𝑥 −5=3 𝑒 −𝑥
( ) 𝑒 𝑥
2 𝑒 𝑥 −5−3 𝑒 −𝑥 =0
2 𝑒 2𝑥 −5 𝑒 𝑥 −3=0
2 𝑒 𝑥 +1 𝑒 𝑥 −3 =0
𝑒 𝑥 =− 1 2
𝑒 𝑥 =3
𝑥= ln − 1 2
𝑥= ln 3
𝑥=1.10

14. Exponential Equations
Solve for 𝑥 to nearest hundredth.
Exponential Equations
𝑒 𝑥 +5 𝑒 −𝑥 −14=0
( ) 𝑒 𝑥
8 𝑒 2𝑥 −14 𝑒 𝑥 +5=0
4 𝑒 𝑥 −5 2 𝑒 𝑥 −1 =0
𝑒 𝑥 = 5 4
𝑒 𝑥 = 1 2
𝑥= ln 5 4
𝑥= ln 1 2
𝑥=0.22
𝑥=−0.69

15. Solve for 𝑥 to nearest hundredth.
Check – up
𝑥 4 𝑥 −5 =3
𝑥=1.45

16. Exponential Equations
Sometimes we can’t solve algebraically, so we go to our graphing calculator.
Exponential Equations
Solve using a graphing calculator.
11. 𝑒 𝑥 = 𝑥 2 −1
𝑌 1 = 𝑒 𝑥
(Find intersection)
𝑌 2 = 𝑥 2 −1
𝑥=−1.15
12. 𝑦≥ 𝑒 𝑥 −2
𝑌 1 = 𝑒 𝑥 −2

17. Compound Interest Applications Compound Interest Formula:
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
𝐴 = total value of investment
𝑡 = number of years
𝑃 = principal amount invested
𝑟 = interest rate (%  decimal)
𝑛 = number of times per year interest is compounded

18. Compound Interest 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴 = 20,000 𝑟 = 0.083 𝑛 = 12 𝑃 = 11,500
13. The Smith Family wants to give their youngest daughter $20,000 when she is ready for college. They now have $11,500 to invest. Determine how many years it will take them to achieve their goal given that they invest this amount at 8.3% compounded monthly.
Compound Interest
𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡
𝐴 = 20,000
𝑟 = 0.083
𝑃 = 11,500
𝑛 = 12
𝑡 = ?
20,000=11, 𝑡
20,000 11,500 = 𝑡
log log
log 20,000 − log 11,500 =12𝑡 log
𝑡= log 20,000 − log 11, log
𝑡=7 𝑦𝑒𝑎𝑟𝑠
*Watch those parentheses!

19. Compound Interest Continuous Compound Interest Formula 𝐴=𝑃 𝑒 𝑟𝑡
14. A sum of money invested at a fixed interest rate, compounded continuously, tripled in 19 years. Determine the interest rate at which the money was invested.
You don’t know A or P but you don’t need it!
You need P to triple
𝐴=𝑃 𝑒 𝑟𝑡
ln 3 =19𝑟 ln 𝑒
3𝑃=𝑃 𝑒 𝑟𝑡
ln =𝑟
3= 𝑒 𝑟𝑡
𝑟=5.8%

20. 4. You invested $4200 in an account paying 6
4. You invested $4200 in an account paying 6.5% compounded continuously. How long to the nearest year will it take for the money in the account to increase by $900?
Check – up
3 years

21. Assignment
Pg. 472 #1, 3, 5, 9, 13, 18, 20–23, 25, 27, 29, 32, 33, 36, 38, 39–47 odd