Logarithmic FunctionsLogarithmic Functions

ObjectivesObjectives

To write exponential equations in To write exponential equations in logarithmic form.logarithmic form.

To use properties of logarithms to expand To use properties of logarithms to expand and condense logarithmic expressions.and condense logarithmic expressions.

Don’t be afraid of logs!

Logarithmic FunctionsLogarithmic Functions

Key to understanding logarithms:Key to understanding logarithms:

A logarithm is an A logarithm is an exponentexponent!!

logB A CExponent

Base Argument

CB A

Logarithmic FunctionsLogarithmic Functions

32 8 2log 8 3

25 25 5log 25 2

10 7x 10log 7 x

34 64 4log 64 3

5 125x 5log 125 x

Exponential FormExponential Form Logarithmic FormLogarithmic Form

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: 3log 9 n

3 9n

2n

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: 5log 1 n

5 1n

0n

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: 4log 2 n

4 2n

2 12 2n

2 12 2n

2 1n

12n

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: 5log 5 n

5 5n

1n

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: 35log 5 n

35 5n

3n

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: 10log 1000 n

10 1000n

3n

Logarithmic FunctionsLogarithmic Functions

Evaluate:Evaluate: log0.01 n

110010n

21

1010n

210 10n

2n

Special BasesSpecial Bases

10log logA A

log lne A A

Common log

Natural log

Natural LogarithmNatural Logarithm

Evaluate:Evaluate: ln1

lne

4ln e

0

1

4

Properties of LogarithmsProperties of Logarithms

ln ln lnAB A B

ln ln lnA

A BB

ln lnBA B A

ln Ae A

ln Ae A

Properties of LogarithmsProperties of Logarithms

Expand:Expand:23

lnx

y

2ln3 lnx y2ln3 ln lnx y

ln3 2ln lnx y

Properties of LogarithmsProperties of Logarithms

Expand:Expand: 2ln 1x x

2ln ln 1x x

ln 2ln 1x x

ln does not distribute!

ln 1 ln ln1x x

Properties of LogarithmsProperties of Logarithms

Expand:Expand:2

3ln

6

x

y

2 3ln ln 6x y

2 3ln ln 6 lnx y

2ln ln 6 3lnx y

Properties of LogarithmsProperties of Logarithms

Expand:Expand: 2

3ln xy

1

22

3ln xy

2

312 ln x

y

2 312 ln lnx y

12 2ln 3lnx y

32ln lnx y

Properties of LogarithmsProperties of Logarithms

Combine:Combine: ln 4 ln x

4ln x

Properties of LogarithmsProperties of Logarithms

Combine:Combine: 2ln8 5ln z

2 5ln8 ln z

5ln 64z

Properties of LogarithmsProperties of Logarithms

Combine:Combine: ln 1 ln 2 3lnx x x

3ln 1 2 lnx x x

2 3ln 3 2 lnx x x

2

33 2ln x xx

Properties of LogarithmsProperties of Logarithms

Combine:Combine: 4ln3 2ln lnx y

4 2ln3 ln lnx y

281ln lnx

y

281lnx y

Properties of LogarithmsProperties of Logarithms

Combine:Combine: 2122ln3 ln 1x

1

22 2ln3 ln 1x

2

9

1ln

x

Solving Equations Using LogsSolving Equations Using Logs

Solve:Solve: Solve:Solve:24 64x 2 34 4x 2 3x

2 7x

ln 2 ln 7x

ln 2 ln 7x 1x ln 7

ln 2x

2.807x

Solving Equations Using LogsSolving Equations Using Logs

Solve:Solve: Solve:Solve:34 9x 3ln 4 ln9x

( 3)ln 4 ln9x

2 10xe

5xe

ln ln5xe ln9ln 43x

ln9ln 4 3x

4.585x

ln5x

1.609x

Solving Equations Using LogsSolving Equations Using Logs

Solve:Solve: Solve:Solve:2 15 2 115xe 2 12 110xe 2 1 55xe

32 1.5 640x

1.5 20x

ln 1.5 ln 20x

2 1ln ln55xe 2 1 ln55x

ln 1.5 ln 20x ln 20ln1.5x

2 ln55 1x

1.504x

7.39xln55 12x

Solving Equations Using LogsSolving Equations Using Logs

Solve:Solve: Solve:Solve: 250 3 125xe 23 2.5xe

2 0.5xe

8ln 3 2 1.5x ln(3 2) 0.1875x

0.1875 3 2e x 2 0.5xe 2ln ln 0.5xe 2 ln 0.5x

0.1875 2 3e x 0.1875 2

3ex

ln 0.52x

0.35x

1.07x

Using LogarithmsUsing Logarithms

Suppose you deposit money into an account whose Suppose you deposit money into an account whose annual interest rate is 4% compounded continuously. annual interest rate is 4% compounded continuously. How long will it take for the money to double?How long will it take for the money to double?

rtA Pe0.042 tP Pe

0.042 te0.04ln 2 ln te

ln 2 0.04t17.3 yearst

Using LogarithmsUsing Logarithms

Chris wants to buy a car. It costs $30,000, and he only Chris wants to buy a car. It costs $30,000, and he only has $27,000. If he invests it in a bank account at 6% has $27,000. If he invests it in a bank account at 6% interest compounded quarterly, how long will he have to interest compounded quarterly, how long will he have to wait before it matures to $30,000?wait before it matures to $30,000?

1ntr

nA P

40.06430000 27000 1

t

4109 1.015

t

4109ln ln 1.015

t109ln 4 ln1.015t

109

ln

ln1.015 4t

1.77 yearst

Logarithmic FunctionsLogarithmic Functions

Consider:Consider: 2( ) logf x x

2(8) log 8f 1 1

24 4( ) logf

2(2) log 2f

2(1) log 1f

3

2

1

0

Logarithmic FunctionsLogarithmic Functions

2( ) logf x x

Logarithmic FunctionsLogarithmic Functions

The inverse of a logarithmic function is an The inverse of a logarithmic function is an exponential function.exponential function.

Logarithmic FunctionsLogarithmic Functions

2( ) logf x x

( ) 2xf x (red)

ConclusionConclusion

A logarithm indicates the exponent to which you A logarithm indicates the exponent to which you raise a certain base in order to produce a given raise a certain base in order to produce a given value.value.

The inverse of logarithmic function is an The inverse of logarithmic function is an exponential function.exponential function.

Logs to the base 10 are written without a base.Logs to the base 10 are written without a base.

Logs to the base Logs to the base ee are indicated by the symbol are indicated by the symbol lnln..