2 3 2 -2 = A. 1 B. 2 C. 3 D. 4. 2 3 2 -2 = A. 1 B. 2 C. 3 D. 4.

223322-2-2 = =

A.A. 11

B.B. 22

C.C. 33

D.D. 44

= =

A.A. 256256

B.B. ¼ ¼

C.C. 1616

D.D. ½ ½

2

1

16

==

A.A. 11

B.B. 22

C.C. 33

24 1

24 1

2 2

2

Logarithmic FunctionLogarithmic Function

DefinitionDefinition

A function f : (0,+oo) -> R is a A function f : (0,+oo) -> R is a logarithmic function means that logarithmic function means that

1) a positive b, b 1, and1) a positive b, b 1, and

2) f(x) = 2) f(x) =

ExamplesExamples

loglog2216 = 4 because 216 = 4 because 244 = 16 = 16



LogLog22(1/16) = -4 because 2(1/16) = -4 because 2-4-4 = 1/16 = 1/16

loglogeeee33 = 3 because e = 3 because e33 = e = e33

loglog4416 = 16 =

A.A. 11

B.B. 22

C.C. 33

D.D. 44

loglog555 = 5 =

A.A. 11

B.B. 22

C.C. 33

D.D. 44

LogLog22[ ¼ ] = [ ¼ ] =

A.A. -1-1

B.B. -2-2

C.C. -3-3

D.D. -- 2

ExamplesExamples

1. f(x) = log1. f(x) = log22(x)(x)

2. f(x) = log2. f(x) = log½½(x)(x)

ExamplesExamples

1. f(x) = log1. f(x) = log22(2(2xx) = x) = x

2. f(x) =2. f(x) =

1

2

log1

2

x

x

ExamplesExamplesDomain = Domain =

Range =Range =Passes through (1,0)Passes through (1,0)Increasing if b>1Increasing if b>1Decreasing if 0<b<1Decreasing if 0<b<1Passes through (b,1)Passes through (b,1)Continuous everywhereContinuous everywhereDifferentiable everywhereDifferentiable everywhere

, R

0,

Theorem on logarithmsTheorem on logarithms

log(xy) = log(x) + log(y) log(xy) = log(x) + log(y) log(x/y) = log(x) – log(y)log(x/y) = log(x) – log(y) log(xlog(xpp) = p log(x)) = p log(x) loglogbb(b(bxx) = x because log and exponent ) = x because log and exponent

killkill

bbloglogbb(x)(x) = x because exponent and log kill = x because exponent and log kill


log(xy) = log(x) + log(y) log(xy) = log(x) + log(y) log(xlog(xpp) = p log(x)) = p log(x)

log x(x+1)log x(x+1)4 4 = log(x) + log(x+1)= log(x) + log(x+1)44..

= log(x) + 4log(x+1)= log(x) + 4log(x+1)

loglog

A.A. loglog44(x(x22+1)+log+1)+log44xx55

B.B. loglog44(x(x22)+log)+log44(1)+log(1)+log44xx55

C.C. [log[log44(x(x22)+log)+log44(1)]log(1)]log44xx55

2 54 ( 1)x x


log(xy) = log(x) + log(y) log(xy) = log(x) + log(y) log(xlog(xpp) = p log(x)) = p log(x)

log x(x+1)log x(x+1)4 4 = log(x) + log(x+1)= log(x) + log(x+1)44..

= log(x) + 4log(x+1)= log(x) + 4log(x+1)

loglog44(x(x22+1)+log+1)+log44xx55 = =

A.A. 5log5log44(x(x22+1)+log+1)+log44xx55

B.B. loglog44(x(x22+1)+log+1)+log445x5x44

C.C. loglog44(x(x22+1)+5log+1)+5log44xx

Natural logarithmic fnNatural logarithmic fn

f(x) = ln(x)f(x) = ln(x)

#41 Solve for t.#41 Solve for t.

50/(1+4e50/(1+4e0.2t0.2t) = 20) = 20

cross multiplycross multiply

20(1+4e20(1+4e0.2t0.2t) = 50) = 50

0.2

50 20

1 4 1te


20(1+4e20(1+4e0.2t0.2t) = 50) = 50

(1+4e(1+4e0.2t0.2t) = 50 / 20 = 2.5) = 50 / 20 = 2.5

4e4e0.2t0.2t = 1.5 divide both sides by = 1.5 divide both sides by 44

ee0.2t0.2t = 0.375 = 0.375


ee0.2t0.2t = 0.375 isolate e and take ln of both sides = 0.375 isolate e and take ln of both sides

ln(eln(e0.2t0.2t) = ln(0.375) ln kills e) = ln(0.375) ln kills e

0.2t = ln(0.375) divide both sides by 0.20.2t = ln(0.375) divide both sides by 0.2

t = ln(0.375) / 0.2 = -0.98/0.2t = ln(0.375) / 0.2 = -0.98/0.2

= - 4.90= - 4.90

ExamplesExamples

1. f(x) = e1. f(x) = exx

2. f(x) = (1/e)2. f(x) = (1/e)x x = e= e-x-x

TrackingTrackingGPSGPS

0.6 [0.6 [22ndnd][ln] 0.17 * 2 [)] = 0.843][ln] 0.17 * 2 [)] = 0.843

#38 Estimated number of automatic #38 Estimated number of automatic vehicle trackers in US is vehicle trackers in US is

where t=0 corresponds to year where t=0 corresponds to year 2000 and N(t) in millions2000 and N(t) in millions

0.17( ) 0.6 tN t e( ) xf x e

( )N t

0.17(2)(2) 0.6 0.843N e xe


a)a) What was the number What was the number installed in 2000?installed in 2000?

0.6 [0.6 [22ndnd][ln] 0.17 * 0 [)] = 0.6][ln] 0.17 * 0 [)] = 0.6

0.17( ) 0.6 tN t e

xe

N(t) = 0.6 e N(t) = 0.6 e 0.17t0.17t in $10 in $1066

How many million in How many million in 2005?2005?

A.A. 1.11.1

B.B. 1.21.2

C.C. 1.41.4


0.6 [0.6 [22ndnd][ln] 0.17 * 5 [)] = 1.404][ln] 0.17 * 5 [)] = 1.404

How many were installed in How many were installed in 2005?2005?

0.17( ) 0.6 tN t exe

ExamplesExamples

f(x) = (1/e)f(x) = (1/e)x x = e= e-x-x

The disability rate in The disability rate in percent for over 65 yrs. percent for over 65 yrs. oldoldt=0 is 1982t=0 is 1982

What is the disability rate What is the disability rate inin

1982?1982?

2000? (-)2000? (-)

26.3 [26.3 [22ndnd][ln] -.016*18[)]][ln] -.016*18[)]

xe

t=0 is 1982. What was t=0 is 1982. What was the disability rate in the disability rate in 1990?1990?

t=0 is 1982. What was t=0 is 1982. What was the disability rate in the disability rate in 1990?1990?

23.1423.14

0.10.1

2 3 2 -2 = A. 1 B. 2 C. 3 D. 4. 2 3 2 -2 = A. 1 B. 2 C. 3 D. 4.

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