Evaluating logarithms

A logarithm is an exponent…if there is no base listed, the common log base is 10.

Evaluate,Using a Calculator:

log 100

log 1000

log5 25

Evaluating logarithms

A logarithm is an exponent…if there is no base listed, the common log base is 10.

Evaluate: log10100 =2 → 102 =100

log101000 =3→ 103 =1000

log5 25 =2 → 52 =25

Using the calculator-evaluate to the nearest hundredth

log 20

log .01

Using the calculator

log 20 ≈1.30

log .01=−2

The natural logarithm: ln

The natural log is base e

log10=1log5 5 =1log2 2 =1lne=1

The natural logarithm: ln

The natural log is base e

log1010=1log5 5 =1log2 2 =1lnee=1

Logarithmic exponential

Log base value = exponent

Log 2 8 = 3 23 = 8

Solve for x:Log 6 x = 3

Logarithmic exponential

Log base value = exponent

Log 2 8 = 3 23 = 8

Solve for x:Log 6 x = 3 63 = x

x = 216

Change of base law:

When not a common logarithm use the change of baseLaw:

log3 27

log4 20

logb c=log clog b

Change of base law:

When not a common logarithm use the change of baseLaw:

log3 27=log 27log 3

=3

log4 20 =log 20log 4

≈2.16

logb c=log clog b

Numerical example-evaluate:

log2 32

Laws of Logarithms

log ab=loga+ logb

logab

=loga−logb

log ac =cloga

Example: from previous slide..

log2 32=5log2 32=log2 (16 • 2)But this is

also true:

Example:

Examples: expand using log laws:

loga3b

log(xy)4

logb2

c

log a

log ab=loga+ logb

logab

=loga−logb

log ac =cloga

Examples:

loga3b=loga3 + logb=3loga+ logb

log(xy)4 =4 log(xy) =4 logx+ 4 logy

logb2

c=logb2 −logc=2 logb−logc

log a=loga12 =

12loga

Single logarithms

log2−3logx+12logy

Single logarithmsdivision, then multiplication:

log2−3logx+12logy

log2

x3y12

log2

x3 y

Express as a single logarithm:

3loga+ logb2 logb−logc13log f

Express as a single logarithm:

3loga+ logb=loga3b

2 logb−logc=log(b2

c)

13log f = f3

Expansion with substitutionlog x =a logy=b

logx3y

First expand then substitute

Given:

Find:

Expansion with substitutionlog x =a logy=b

logx3y

First expand then substitute

3log x + logy3a+b

With numbers….log9 4 =a log9 5 =blog9 20 =

log9100 =

Rewrite the numbers in terms of factors of 4 and 5 only

With numbers….log9 4 =a log9 5 =blog9 20 =log9 (4 • 5) =log9 4 + log9 5 =a+b

log9100 =log(52 • 4) =2 log5 + log4 =2b+a

Rewrite the numbers in terms of factors of 4 and 5And use and find log9 9=1

log99

16

solution

log99

16=log9

942

=log9 9 −2 log9 4 =1−2a

Try page 335 46, 50,52

Solving log equations: the domain of y = log x is that x>0!

x1

3 =5

Do Now:1. solve the exponential equation:

2. Solve the logarithmic equation:

log4 x =12

Solving log equations when the base is a variable:

x1

3 =5Recall to raise both change to an sides to the reciprocal power: exponential equation:

(x1

3 )3 =53

x=125

log4 x =12

412 =xx=2

log4 x =12

Solving log equations….(combine the last 2 concepts)

logx 64 =32

1. Change to an exponential equation2. Raise to the reciprocal power.3. Check all answers!! (Solutions may be extraneous)

Solving log equations….

logx 64 =32

x32 =64

(x32 )

23 =64

23

x=16

1. Change to an exponential equation2. Raise to the reciprocal power.

power

root

More examples: in ex. 1, note that the domain is: x-2>0 so…x>2

log3(x−2)=2

logx13

=−1

Solutions:log3(x−2)=2

32 =x−2x=11

logx13

=−1

x−1 =13

x=3

1.

2.

Express as a single log first!

log2+2 logx=3log2

Express as a single log first!

log2+2 logx=3log2

log2 + logx2 =log23

2x2 =8x=±2{2}

-2 is extraneous!

When you can’t rewrite using the same base, you can solve by taking

a log of both sides

2x = 7log 2x = log 7x log 2 = log 7

x = ≈ 2.8072log

7log

Example: 4x = 15

Example: 4x = 15

log 4x = log 15

x log 4 = log15

x= log 15/log 4

≈ 1.95

Using logarithms to solve exponentials

ex =200

Here we should “ln” both sides….

Using logarithms to solve exponentials

ex =200

lnex =ln200xlne=ln200

x=ln200lne

x≈5.298

Solving with logs-isolate first…

4(3)x =3200

Solving with logs-isolate first…

4(3)x =3200

3x =800

log3x =log800xlog3=log800

x=log800log3

x≈6.085

Isolate the base term first!

102x +4 = 21

Isolate the base term first!

102x +4 = 21102x = 17

log 102x=log 17

Isolate the base term first!

102x +4 = 21102x = 17

log 102x=log 17

2xlog 10 = log 17

2x log10

(2 log10)=

log17(2 log10)

x≈.6152Use ( )!

Graphs of exponentials

• Growth and decay:

• growth decay

Compound Formula

• Interest rate formula

A=a0 1+rn

⎛

⎝⎜

⎞

⎠⎟nt

Compound Formula

• How long will it take $200 to become $250 at 5% interest rate, compounded quarterly

A=a0 1+rn

⎛

⎝⎜

⎞

⎠⎟nt

250=200(1+.054

)4t

Compound Formula

• How long will it take 200 to become 250 at 5% interest rate, compounded quarterly

250=200(1+.054

)4t

250200

=200200

(1.0125)4t

1.25 =1.01254t

solution

• Log both sides and round to the nearest year

250=200(1+.054

)4t

1.25 =1.01254t

log1.25 =4tlog1.0125log1.25

(4 log1.0125)=t

x≈4 years

CONTINUOUS growth:

Ex: population grows continuously at a rate of 2% in Allentown. If Allentown has 10,000 people today, how many years will it takeTo have about 11,000 to the nearest tenth of a year?

A=a0ert

A=a0ert

11,000 =10,000e(.02x)

11,00010,000

=10,00010,000

e.02x

CONTINUOUS growth:

A=a0ert

11,000 =10,000e(.02x)

11,00010,000

=e.02x

1.1=e.02x

ln1.1=.02xlneln1.1

(.02 lne)=x

x≈4.8 years

Solving Log Equations

• To solve use the property for logs w/ the same base:

• If logbx = logby, then x = y

log3(5x-1) = log3(x+7)

Solve by decompressing

log3(5x-1) = log3(x+7)•5x – 1 = x + 7• 5x = x + 8• 4x = 8• x = 2 and check• log3(5*2-1) = log3(2+7)• log39 = log39

Example:

• Solve:

3log x =log27

Example:

• Solve:

3log x =log27

x3 =27

x=2713

x=3

log5x + log(x+1)=log100• Decompress

log5x + log(x+1)=log100• (5x)(x+1) = 100 (product property)

• (5x2 + 5x) = 100 5x2 + 5x-100 = 0• x2 + x - 20 = 0 (subtract 100 and divide by 5)

• (x+5)(x-4) = 0 x=-5, x=4• 4=x is the only solution

another

• Solve: log x−log7 =log21

another

• Solve:log x−log7 =log21x7

=21

x=147

One More!log2x + log2(x-7) = 3

Solve and check:

One More!log2x + log2(x-7) = 3

• log2x(x-7) = 3• log2 (x2- 7x) = 3• 2log

2(x -7x) = 23

• x2 – 7x = 8• x2 – 7x – 8 = 0• (x-8)(x+1)=0• x=8 x= -1

2

Graphs of exponentials

• Growth and decay:

• growth decay

Inverse functions• Inverse functions are a reflection in y=x

Y=2x

Y=log2x

Y=x

Domain of y=2x is all realsDomain of y = log2x is (0,∞)