Logarithms and Exponential
Equations
Ashley Berens
Madison Vaughn
Jesse Walker
Logarithms
• Definition- The
exponent of the
power to which a
base number must
be raised to equal a
given number.
Evaluating logarithms
• If b > 0, b ≠ 1, and x >0 then…
Logarithmic Form
log b x = y
• Examples…
– log 81 = x
3
x
3 = 81
x=4
….
Exponential Form
by = x
x
– 2 =2
x=1
Basic Properties
•
logь1=0
• logьb=1
x
• logьb =x
• ьLogbx=x, x>0
]-
Inverse properties
Examples of Basic Properties
• Log 5125 =
5x = 125
x=3
• 12
• log 9 81=
9x = 81
x=2
• 3
log
4.7
12
12 log 124.7
( 12’s Cancel)
Log = 4.7
log
3
3
1
log
3
1
( 3’s Cancel)
Log = 1
Common Logarithms
• If x is a real number then the following is
true…
•
•
•
•
Log 1 = 0
Log 10x = 1
Log 10 = x
log x
10
= x, x > 0
]-
Inverse Properties
Common Logs
• Log 0.001
log 1/ 1000 = log
-3 = log 10 -3
log = -3
1/103
• Log(-5)
10 x = -5
NO SOLUTION
( Because it’s a
negative)
• Log -0
x
10 = 0
NO SOLUTION
• Log 10,000
x
10 = 10,000
x=4
Natural logs
• If x is a real number then….
•
•
•
•
ln 1 = 0
ln e = 1
x
ln e = x
ln x
e = x, x > 0
]-
Inverse properties
Natural log examples
• ln e
ln = 0.73
0.73
• ln ( -5)
No Solution
( Cant have a natural long of a
negative)
• ln 32
e x= 32
x = (Use Calculator)
• e ln 6
e=6
Expanding Logarithms
• log12x 5 y -2
= log12 logx 5+ logy -2
= log12 + 5logx – 2logy
• ln
X
2
√4x+1
= lnx2 - ln √4x+1
= 2lnx – ½ ln (4x+1)
Condensing logarithms
• -5 log (x+1) + 3 log (6x)
2
2
= 3log (6x) – 5log (x+1)
2
2
= log 2 6x - log 2(x+1)5
(6a)3
= log 2
(x+1) 2
Change of base
•For any positive real numbers a, b and x, a ≠1 , b ≠1
• log 5
3
= log5
log3
• log 4212
78
(Use Calculator)
=1.34649…
• log½6
= log6
log ½
(Use Calculator)
= -2.5849…
= log 4212
log 78
= (Use Calculator)
= 1.9155…
• log 33
15
= log 33
log 15
= (Use Calculator)
= 1.2911…
Exponential Functions
 Exponential functions are of the form
x
f(x)=ab, where a≠0, b is positive and
b≠1. For natural base exponential
functions, the base is the constant e.
 If a principle P is invested at an annual
rate r (in decimal from), then the
balance A in the account after t years is
given by:

Formulas
•
nt
A = P( 1+r/n )
•
•
When compounded n
times in a year.
A = Pe
•
rt
When compounded
continuously.
ExponEntial ExamplEs…
•
New York has a population of approximately 110 million. Is New York's population continues at the
described rate, predict the population of New York in 10…
–
A. 1.42% annually
t
F(x) = 110 * (1+ .0142)
F(x)= 110 * 1.0142
t
F(10) = 126,657,000
–
B. 1.42% Continuously
N = Pe rt
(.0142 * t)
N(t) = 110e
N(t) = 126,783,000
Finding growth and decay
• 562.23 * 1.0236
t
•If the number is more than one
than it is an exponential increase.
•If it is less than one than it is a
exponential decrease.
<- Exponential Growth