The Natural Base, e HW
Evaluate each expression to the nearest thousandth. If the expression is undefined, write undefined.
1. 𝑒 9
2. 𝑒 3.4
3. 3𝑒 0.05
4. 3𝑒 −0.257
1
5. 𝑒 4
6. ln 7
9. ln √5
10. ln (-3)
7. ln 99,999
Write in ascending order.
1
11. 𝑒, 𝑒 0 , ln 1, ln 2
8. ln 0.994
12. 𝑒 1.3 , ln 1.3, 101.3 , log 1.3
State whether each equation is always true, sometimes true, or never true.
13. 𝑒 5𝑥 ∙ 𝑒 3 = 𝑒 15𝑥
Simplify the expression.
15. 𝑒 ln 5
14.
16. 𝑒 2 ln 5
Write an equivalent exponential or logarithmic equation.
19. 𝑒 𝑥 = 1
20. ln 5 ≈ 1.61
𝑒 8𝑥
𝑒4
= 𝑒 2𝑥
17. ln 𝑒 4
18. 2 ln 𝑒 4
21. 𝑒 0.69 ≈ 1.99
Solve each equation for x by using the natural logarithm function. Round your answers to the nearest hundredth.
22. 1.3x = 8
23. 362x = 20
1
24. 2−3𝑥 = 10
Challenge:
25. Sketch 𝑓(𝑥) = 𝑒 𝑥 for −1 ≤ 𝑥 ≤ 2. A line that intersects a curve at only one point is called a tangent line of the
curve.
a. Sketch lines that are tangent to the graph of 𝑓(𝑥) = 𝑒 𝑥 at x = 0.5, x = 0, x = 1, and x = 2.
b. Find the approximate slope of each tangent line. Compare the slope of each tangent line with the
corresponding y-coordinate of the point where the tangent line intersects the graph.
c. Make a conjecture about the slope of 𝑓(𝑥) = 𝑒 𝑥 as x increases.
Let 𝒇(𝒙) = 𝒆𝒙 . For each function, describe the transformations from f to g.
26. 𝑔(𝑥) = 6𝑒 𝑥 + 1
27. 𝑔(𝑥) = 0.25𝑒 (4𝑥+4)
Let 𝒇(𝒙) = 𝒍𝒏 𝒙. For each function, describe the transformations from f to g.
28. 𝑔(𝑥) = 3 ln(𝑥 + 1)
29. 𝑔(𝑥) = −2 ln(𝑥 − 1)
30. The graphs of 𝑓(𝑥) = 𝑒 −2𝑥 , 𝑔(𝑥) = 𝑒 −𝑥 , ℎ(𝑥) = 𝑒 𝑥 , 𝑎𝑛𝑑 𝑖(𝑥) = 𝑒 2𝑥
are shown on the same coordinate plane to the left. What transformations
relate each function, 𝑓, 𝑔, 𝑎𝑛𝑑 𝑖, 𝑡𝑜 ℎ?
31. For 𝑓(𝑥) = 𝑙𝑛 𝑥, describe how each transformation affects the domain, range, asymptotes, and x-intercepts.
a. vertical stretch
b. horizontal stretch
c. a vertical translation
d. a horizontal translation
32. The factory sales of pagers from 1990 to 1995 can be modeled by the functions S = 116𝑒 0.18𝑡 , where t = 0 in 1990
and S represents the sales in millions of dollars. [Source: Electronic Market Data Book]
a. According to this function, find the factory sales of pagers in 1995 to the nearest million.
b. If the sales of pagers continued to increase at the same rate, when would the sales be double the 1995
amount?
33. A wooden chest is found and is said to be from a second century B.C.E. Tests on a sample of wood from the chest
reveal that it contains 92% of its original carbon-14. Could the chest be from the second century B.C.E.?
Assume that all interest rates are compounded continuously.
34. How long will it take an investment of $5000 to double if the annual interest rate is 6%?
35. If it takes a certain amount of money 3.7 years to double, at what annual interest rate was the money invested?