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