Exponential and Logarithmic Equations
Warm Up
1. You bought a new boat in 2013. The boat was $12,000 and the value depreciates 15% per year. In what year will the car’s value be $5,000? 2. Suppose $18,000 is deposited into a savings account paying 6%, compounded monthly. How much money is in the account 5 years? 3. If $1700 is deposited into an account paying 3.5% annual interest, compounded continuously, how much is in the account after 12 years? 5. Evaluate the logarithm 1
64
7. Expand using log properties. log ! 𝑥 ! 𝑦 !
log !
4. The way a population of a city changes, represented in millions, is given by the equation 𝑦 = 13(0.87)! . What percentage is the population changing by? 6. Write as a single logarithm. 2log ! 3𝑥 − log ! 6 8. Find the inverse: f(x)= log6 3t +7 Facts about the number e and the natural log…
ü The number e is called the _______________________. ü For our purposes, it has been used to show __________________________ growth. § Continuous compound interest formula: ü e is an irrational number that is approximately equal to _______________. Convert 𝑒 ! = 𝑥 to logarithmic form. ü e is the base of __________________________, denoted ____________. ü FACT: ln 𝑒 = _______ Write an equivalent exponential or logarithmic equation.
1. ln 50 = x 2. ln 36 = 2x 3. ex = 8 4. e5 = 10x Evaluate each expression. 5.
eln12 7. ln e −1 6. eln 3x 8. ln e−2 y Solving Exponential and Logarithmic Equations ü Exponential Equations (common base type) – create common base and set exponents equal.
1. 2x+4 = 25
2. 76 x = 72 x−20
3. 22 x+2 = 64
4. 22 x+3 = 32
5. 43 x−2 =16
6. 32 x +5 = 27 x
ü Classic Exponential Equations (classic exponential type) – isolate exponential base and convert to log form.
7. 7y + 2 = 17
8. e3 x = 8
10. 53b = 106
11. 2(92m )= 54
9. 3r − 5 = 4.1
12. e3 x −5 = 32
ü Logarithmic Equations (log on a single side type) – isolate one log and onvert to exponential form. Use log properties if necessary.
13. log 2𝑥 = −1
14. 3log(3𝑥 + 1) = 6
15. log ! 8𝑥 − 3 = 3
16. ln 2 + ln x = 3
17. log4 x – 3 log4 2 = 2
18. log3 y – log3 (2 − y) = 0
ü Logarithmic Equations (log on both sides type) – if log ! 𝑥 = log ! 𝑦, then 𝑥 = 𝑦. 19. log2 q + log2 3 = log2 30
20. log10 27 = 3 log10 x
21. ln 36 − ln x = ln 6
22. log9 (3u + 14) − log9 5 = log9 2u
23. log3 5x + log3 4 = log3 140
* 24. log4 (2x+1) = log4 (x-3) + log4 (x+5)
Practice!
Solve the following exponential or logarithmic equations. 9. e
0.5 x
=6
10. ln 4x = 3 11. ln 6 + ln x = 4
12. ln 2.5x = 10