Summary of Chapter 7
Name: ____________________________________
Use pages 343 – 345 to help you answer the following.
1. The standard form of an exponential function is ____________________.
The function models exponential growth, when _____________________.
The function models exponential decay, when ______________________.
Function
2.
Y = 15(1.325)x
3.
Y = 5(0.35)x
Growth or Decay?
% of growth or decay
Initial Value (y-intercept)
Write an exponential function to model each situation.
4. A home worth $500,000 appreciates 3% each year.
________________________
5. A car worth $30,000 depreciates 8% each year.
________________________
The formula for continuously compounded interest is A = P ert.
P represents ______________________; r represents ____________________; t represents _____________
6. Find the amount in the account if $1000 is invested for 15 years at an interest rate of 6%
compounded continuously.
____________________________________
The half-life of a substance is the time it takes for _________________________________________________.
In every half-life problem the base of the exponential function is __________.
7. Write an equation to represent the following situation and answer the question.
20 mg of Hg-197, which may be used in kidney scans and has a half-life of 64.1 hours has been
administered. Find the amount remaining after 72 hours.
Equation: _____________________
Answer: ____________________________
Write an equation to describe the exponential function with the given base whose graph passes through the
given point.
8. The base is 4 and the graph passes through (1, 5).
9. The base is ½ and the graph passes through the point (1, 4).
Write each exponential equation in logarithmic form.
10. 52 = 25 _____________________
11. 6x = 216 ___________________________
12. ex = 12 _____________________
13. ex = 25 ___________________________
Write each logarithmic equation as an exponential equation.
14. log232 = 5 __________________
15. log 100 = 2 ________________________
16. ln 20 = 3 ___________________
17. ln 22026.5 = 10 _____________________
Properties of Logarithms
For any positive numbers M, N and b, b ≠ 1, expand each of the following.
18. logbMN = __________________________________________ Product Property
𝑀
19. logb 𝑁 = _________________________________________ Quotient Property
20. logbMx = _________________________________________ Power Property (bring down property)
When you condense a logarithmic expression, you rewrite it as a _____________________________.
When you expand a logarithmic expression, you rewrite it as a ______________________________________.
Write as a single logarithm.
21. log53 + log5x = _______________
22. log 5 + 2log x = __________________________
23. ln 2 – ln x = _______________
24. ln x – 3ln y = __________________________
Expand each logarithmic expression.
25. log3x2y4 = ______________________________________________________________
𝑥
26. ln (2) = ______________________________________________________________
Solving Exponential and Logarithmic Equations
Helpful hint:
If the exponent contains a variable, then take the log of both sides and use the bring down property.
Ex: 5x = 125
Ex:
e3x = 15
Ex: 5(x – 2) = 625
log(5x) = log(125)
ln(e3x) = ln(15)
log(5(x – 2)) = log(625)
x log(5) = log(125)
3x = ln(15)
(x – 2) log(5) = log(625)
x = log(125)
log(5)
x = ln(15)
3
x – 2 = log(625)
log(5)
x=3
x ≈ 0.903
x–2=4
x=6
Solving Exponential and Logarithmic Equations
Helpful hint:
When necessary, condense the logarithmic expression. Convert the equation to exponential form and solve.
Ex: log3 5x = 4
Ex: ln 3 + ln x = 2
Ex: log x + log (x + 1) = 2
34 = 5x
ln 3x = 2
log x(x + 1) = 2
81 = 5x
e2 = 3x
16.2 = x
e2 = x
3
2.463 ≈ x
102 = x(x + 1)
100 = x2 + x
0 = x2 + x – 100
x = – 1 ± √12 − 4(1)(−100)
2
2
x = – 1 ± √1 + 400
2
x = – 1 ± √401
2
x ≈ 9.51 or x ≈ – 10.51
(this answer must
be rejected
because you
can’t take the
log of a negative
number.)
Solve.
2
27. 𝑥 3 = 49
28. log 4x = 1
31. e3x = 15
32. ln x + ln (x + 2) = 10
29. 3 log x – log 2x = 2
33. 5e(x – 2) = 125
30. 2(x – 3) = 8
34. 2 ln x = 12