Download Review for Test #4- Exponential and Log Functions

Exponential & Logarithmic Functions
Definitions, Properties & Formulas
Properties of Exponents
Property
Definition
Product
x a xb  x a  b
Quotient
xa
 x a  b , where x  0
b
x
Power Raised to a Power
(xa)b = xab
Product Raised to a Power
(xy)a = xa ya
Quotient Raised to a Power
x
xa
   a , where y  0
y
y
Zero Power
x0 = 1, where x  0
a
Negative Power
x n 
1
, where x  0
xn
1
n
Rational Exponent
x n x
for any real number x  0 and any integer n > 1
and when x < 0 and n is odd
N = N0 (1 + r)t
Exponential
Growth/Decay
Compound
Interest (Periodic)
Exponential
Growth/Decay
(in terms of e)
Continuously
Compounded
Interest
where: N is the final amount, N0 is the initial amount, t is the number of time
periods, and r is the average rate of growth(positive) or decay(negative) per
time period
r

A  P1  
n

nt
where: A is the final amount, P is the principal investment, r is the annual
interest rate, n is the number of times interest is compounded each year, and t is
the number of years
N = N0 ekt
where: N is the final amount, N0 is the initial amount, t is the number of time
periods, and k (a constant) is the exponential rate of growth(positive) or
decay(negative) per time period
A = Pert
where: A is the final amount, P is the principal investment, r is the annual
interest rate, and t is the number of years
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Logarithmic
Functions
are inverses of exponential functions
 a logarithm is an exponent!
when no base is indicated, the base is assumed to be 10
log x  log10 x
Common
Logarithms
 log x  y  10 y  x
Change of Base
Formula
loga n 
logb n
logb a
where a, b, and n are positive numbers, and a 1, b 1
instead of log, ln is used; these logarithms have a base of e
Natural
Logarithms
ln x  loge x
 ln x = y  e y  x
all properties of logarithms also hold for natural logarithms
Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
Definition
Examples
logb 1  0
written exponentially: b0 = 1
logb b  1
written exponentially: b1 = b
logb b x  x
written exponentially: bx = bx
blog b x  x , where x > 0
10log 10 7  7
logb MN  logb M  logb N
log3 9x  log3 9  log3 x
log 1 yz  log 1 y  log 1 z
5
logb
2
 log 4 2  log 4 5
5
7
log 8  log 8 7  log 8 x
x
log2 6 x  x log2 6
logb M  p logb M
p
if and only if
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log 4
M
 logb M  logb N
N
logb M  logb N
5
log 5 y 4  4 log 5 y
M=N
log6 (3x  4)  log6 (5x  2)
 ( 3 x  4 )  ( 5 x  2)
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Properties of Logarithmic Functions
If b, M, and N are positive real numbers, b  1, and p and x are real numbers, then:
Definition
Examples
logb 1  0
written exponentially: b0 = 1
logb b  1
written exponentially: b1 = b
logb b x  x
written exponentially: bx = bx
blog b x  x , where x > 0
10log 10 7  7
logb MN  logb M  logb N
log3 9x  log3 9  log3 x
log 1 yz  log 1 y  log 1 z
5
logb
2
 log 4 2  log 4 5
5
7
log 8  log 8 7  log 8 x
x
log2 6 x  x log2 6
logb M  p logb M
p
if and only if
5
log 4
M
 logb M  logb N
N
logb M  logb N
5
log 5 y 4  4 log 5 y
M=N
log6 (3x  4)  log6 (5x  2)
 ( 3 x  4 )  ( 5 x  2)
Common Errors:
logb M
 logb M  logb N
logb N
logb (M  N)  logb M  logb N
(log b M)  p logb M
p
logb M  logb N  logb
M
N
logb M
cannot be simplified
logb N
logb M  logb N  logb MN
logb (M  N) cannot be simplified
p logb M  logb Mp
(log b M)p cannot be simplified
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College Algebra: Functions and Models
Review- Exponential and Logarithmic Functions part 1
Name:____________________________________
Date:_____________________________________
ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK!
Write each expression in terms of simpler logarithmic forms:
4
5
(1)
5
log b x y
(2)
log b
s
u7
(3)
logb
1
c8
(4)
logb
m 5n 3
p
Given loga n, evaluate each logarithm to four decimal places:
(5)
log3 42
(6)
(7)
log 1 5
log6 0.00098
2
Solve each equation and round answers to four decimal places where necessary:
(8)
log2 x  3
(10) 1000  75e0.5 x
(12) log 7
1
x
49
(9)
log5 4  log5 x  log5 36
(11) log 6 x  2
(13) log x 4 
1
2
(14) 10 x  27.5
(15) log x  log 5  log 2  log( x  3)
(16) log x  log 2  1
(17) log4 x  3
(18) log9 (5  x)  3 log9 2
(19) log 20  log x  1
(20) 2  1.002 4 x
(21) e 25 x  1.25
(22) log( x  10)  log( x  5)  2
(23) log 6 216 
1
log 6 36  log 6 x
2
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College Algebra: Functions and Models
Review- Exponential and Logarithmic Functions part 2
Name:____________________________________
Date:_____________________________________
SHOW ALL WORK:
(1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22
million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he
need for this investment?
(2)
The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially:
a.
Express the number of guppies as a function of the time t.
b.
Use your answer from part (a) to find how many guppies are present after 1 week?
c.
Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies?
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SHOW ALL WORK:
(3)
The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by
i
log
 0.00235 x . What is the intensity, to the nearest tenth, at a depth of 40 feet?
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(4)
Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula
i
D  10 log , where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per
i0
square meter is a standardized sound level. Use this information and formula to find the intensity of the
sound at the concert.
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SHOW ALL WORK:
(5)
How many years, to the nearest year, will it take the world population to double if it grows continuously at
an annual rate of 2%.
(6)
Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If
you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two
banks?
(7)
Determine how much time, to the nearest year, is required for an investment to double in value if interest
is earned at the rate of 5.75% compounded quarterly.
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