Unit 7 Logarithms

• Exponential functions• Logarithmic functions• Using properties of logarithms• Exponential and Logarithmic equations• Exponential and logarithmic models

8.2 Solving exponential equations and inequalities

To solve exponential equations, get the bases equal.

3 81x Solve for x: 2 12 32a

then

Rememberu va a u v

One to one property!

533 x 5 x Bases must be the same

25 1255 xx

52 2

24 xxBONUS!!

Compound Interest Formulas

After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formula:

1) For n compounding per year: 1nt

rA P

n

Find the account balance after 20 years if $100 is placed in an account that pays 1.2% interest compounded twice a month.

If $350,000 is invested at a rate of 5½% per year, find the amount of the investment at the end of 10 years for the following compounding methods:

a) Quarterly b) Monthly

Solving exponential inequalities is similar to equations, make sure the bases are equal. Solve:

243

13 12 x

32

12 2 x

8.3 Logarithmic Functions

• Write exponential functions as logarithms

• Write logarithmic functions as exponential functions

A logarithm function is another way to write an exponential function

log yay x a x

where 0 and 1 and is read as

"y is the logarithm of with base of .

a a

x a

y is the logarithm, a is the base, x is the number

Common log-base 10When we use a common log with base 10, it is not necessary to indicate the base.

Use the log button on the calculator to take use base 10 log of any number.

Natural log-base e

If we use a natural log, we indicate by writing ln instead of log and no base is needed.

e is an irrational number like π. e = 2.718...

Use the ln button on the calculator to take the natural log of any number.

Evaluate using Change of Base

135log20

How do we evaluate logarithms that are not common?

loglog

logb

ab

MM

a Change of base

formula135log20

log6 8

log3 12

Rewrite as a exponential equation:

3log 5 c 3 5c

log4 16 = 2

log3 729 = 6

log8 512 = 3

log16 8 = 3/4

Rewrite as a logarithm:

2 8x 2log 8 x

43 = 64

1251/3 = 5

113 = 1331

163/4 = 8

To find the exact value of a logarithm (or evaluate), we can change the equation to an exponential one.

2log 16

Evaluate:

log3 81

log1/2 256

log13 169

Evaluate:

128log2

8.4 Solving logarithmic equations

2

3log9 x

Solve:

2

5log16 x

5log 3 2x Change to exponential form

log6x = log69 9 x

log3(x2 - 15) = log3 2x

log4(5x-4) > log43x

Solve Base e Equations

Good to know ln ex = x

4 e-2x - 5 = 3Add 5 to both sides

Divide 4 to both sides

Take ln of both sides

New property

Solve: 3 e4x - 12 = 15

Good to know: eln x = x

Solve Natural Logs

3 ln 4x = 24 5 ln 6x = 8

Continuously Compound Interest A = Pert

Joan was born and her parents deposited $2000 into a college savings account paying 4% interest compounded continuously. What would be the balance after 15 years.

8.5 Using Properties of logarithms

• Rewrite logarithms with a different base

• Use properties of logarithms to evaluate or rewrite logarithmic expressions

• Use properties of logarithms to expand or condense logarithmic expressions

• Use logarithmic functions to model real-life problems

Product Propertylogx ab= logx a + logx b

Quotient Property logx a/b = logx a - logx b

Power of Propertieslogb Ax = xlogb A

Simplify:

4log20log 33

yx 22 loglog3

Expand:

y

x5log

4

2 3log r r3log4 2

2

3 3log

y

3log2 3

y

Use log4 2 = .5 to approximate log4 32

Use log4 3 = .7925 to approximate log4 192

Use log5 2 = .4307 to approximate log5 250

Use log37 = 1.77 to approximate log3 49

Use log5 6 = 1.11 to approximate log5 216

5 5Solve: log 1 log 1 2x x

2 2 2 2Solve: log 3 log log 5 log 2x x

Solve

2)1(log)5(log 1010 xx

Real World ApplicationsThe Ph of a substance is defined as the concentration of hydrogen ions [H+] in moles. It is given by the formula pH = log10(1/H+). Find the amount of hydrogen in a liter of acid rain that has a pH of 4.2.

Also we have the conversion: ln yx y e x

Write in log form:

ex = 9

e7 = x

Write in exponential form:

ln x = 2.143

ln 18 = x

6 ln 8 - 2 ln 4

Simplify the expression:

2 ln 5 + 4 ln 2 + ln 5y

3x = 15

To solve exponential equations, get the bases equal.

Solve: we can’t get the bases equal here.

yxyx aa then ;loglog If

6x = 42 Solve:

Key Chapter points:• Exponential functions

and graphs• Logarithmic functions• Properties of logarithms• Exponential and

Logarithmic equations