logarithms and exponential equations ashley berens madison vaughn jesse walker
TRANSCRIPT
Logarithms and Exponential Equations
Ashley BerensMadison 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 …. Exponential Form
log x = y b = xb y
• Examples… – log 81 = x
3 = 81
x = 4
3x
– 2 = 2
x = 1
x
Basic Properties
• logь1=0
• logьb=1
• logьb =x
• ь b =x, x>0
x
Log x ]- Inverse properties
Examples of Basic Properties
• Log 125 =
5 = 125 x = 3
• log 81= 9 = 81 x = 2
5x
9
x
• 12 12 12 12
( 12’s Cancel)
Log = 4.7
• 3 3 1 3 3 1
( 3’s Cancel)
Log = 1
log 4.7
log 4.7//
log
log/ /
Common Logarithms
• If x is a real number then the following is true…
• Log 1 = 0• Log 10 = 1• Log 10 = x• 10 = x, x > 0
x
log x]- Inverse Properties
Common Logs
• Log 0.001 log = log -3 = log 10 log = -3
• Log(-5) 10 = -5 NO SOLUTION ( Because it’s a
negative)
1/ 1000 1/103
-3
• Log -0
10 = 0
NO SOLUTION
• Log 10,000
10 = 10,000
x = 4
x
xx
Natural logs
• If x is a real number then….
• ln 1 = 0
• ln e = 1
• ln e = x
• e = x, x > 0
x
ln x ]- Inverse properties
Natural log examples
• ln e
ln = 0.73
• ln ( -5)
No Solution
( Cant have a natural long of a
negative)
0.73• ln 32
e = 32
x = (Use Calculator)
• e
e = 6
x
ln 6
Expanding Logarithms
• log12x y= log12 logx + logy
= log12 + 5logx – 2logy
• ln
= lnx - ln = 2lnx – ½ ln (4x+1)
5 -2
5 -2
X 2
√4x+1
2 √4x+1
Condensing logarithms
• -5 log (x+1) + 3 log (6x)
= 3log (6x) – 5log (x+1)
= log 6x - log (x+1)5
= log
22
2 2
2 2
2
(6a)3
(x+1) 2
Change of base
• log 5 = log5 log3
(Use Calculator) =1.34649…
• log 6 = log6
log ½
(Use Calculator) = -2.5849…
• log 4212 = log 4212 log 78 = (Use Calculator) = 1.9155…
• log 33 = log 33
log 15 = (Use Calculator) = 1.2911…
3
½
•For any positive real numbers a, b and x, a ≠1 , b ≠1
78
15
Exponential Functions
Exponential functions are of the form 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:
x
Formulas
• A = P( 1+r/n )• When compounded n
times in a year.
• A = Pe • When compounded
continuously.
nt
rt
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
F(x) = 110 * (1+ .0142)
F(x)= 110 * 1.0142
F(10) = 126,657,000
– B. 1.42% Continuously
N = Pe
N(t) = 110e
N(t) = 126,783,000
t
t
rt(.0142 * t)
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