Calculus 2nd Edition Rogawski

362 CHAPTER 7 EXPONENTIAL FUNCTIONS
• Derivative formulas:
(e x ) ′ = e x ,
• Integral formulas:
d
dx ln x = 1 x , (bx ) ′ = (ln b)b x ,
d
dx log b x = 1
(ln b) x
ln x =
∫ x
1
dt
t
(x > 0),
∫ dx
x
= ln |x|+C
7.3 EXERCISES
Preliminary Questions
1. Compute log b 2(b 4 ).
2. When is ln x negative?
3. What is ln(−3)? Explain.
4. Explain the phrase “The logarithm converts multiplication into
addition.”
Exercises
In Exercises 1–16, calculate without using a calculator.
1. log 3 27 2. log 5
1
25
3. ln 1 4. log 5 (5 4 )
5. log 2 (2 5/3 ) 6. log 2 (8 5/3 )
7. log 64 4 8. log 7 (49 2 )
9. log 8 2 + log 4 2 10. log 25 30 + log 25
5
6
11. log 4 48 − log 4 12 12. ln( √ e · e 7/5 )
13. ln(e 3 ) + ln(e 4 ) 14. log 2
4
3 + log 2 24
15. 7 log 7 (29) 16. 8 3 log 8 (2)
17. Write as the natural log of a single expression:
(a) 2ln5+ 3ln4
18. Solve for x: ln(x 2 + 1) − 3lnx = ln(2).
In Exercises 19–24, solve for the unknown.
(b) 5ln(x 1/2 ) + ln(9x)
19. 7e 5t = 100 20. 6e −4t = 2
21. 2 x2 −2x = 8 22. e
2t+1 = 9e
1−t
23. ln(x 4 ) − ln(x 2 ) = 2 24. log 3 y + 3 log 3 (y 2 ) = 14
25. Show, by producing a counterexample, that ln(ab) is not equal to
(ln a)(ln b).
26. What is b if (log b x) ′ = 1
3x ?
27. The population of a city (in millions) at time t (years) is
P (t) = 2.4e 0.06t , where t = 0 is the year 2000. When will the population
double from its size at t = 0?
5. What are the domain and range of ln x?
6. Does x −1 have an antiderivative for x