Logarithmic Functions

Understanding Logarithms

The ____________________ of an exponential function is called a __________________________ function. Example: In order to solve for y, the logarithmic form is used: Exponential form Logarithmic form where c is a postive # other than 1. Since our number system is based on powers of ______, logarithms with base ______

are widely used and are called ____________________ logarithms. When you write a

___________________ logarthim, you do not need to write the base.

Example: Evaluating Logarithms To evaluate, it is usually easiest to convert to _____________________ form and solve. Example 1: Evaluate:

a.) 7log 49 b.) 61log36

c.) 6log 1

d.) 9log 27 e.) log 0.001 f.) 2log 8 g.) log100 Example 2: Solve for x:

a.) 5log 3x = − b.) 642log3

x = c.) 3log 84x =

d.) log 36 2x = e.) ( )2 3log log 9 x=

Graphing Sketch: 3xy = and the inverse of 3xy =

Note:

Example 3: Sketch 2logy x=

y

x

Domain: Increasing or decreasing?

Range: Equations of asymptote:

Example 3: Sketch 13

( ) logf x x=

Domain:

Range: Equations of asymptote:

Increasing or decreasing?

Example 4: Estimate the value of 2log 14 . Justify your answer.

Rather than estimating, you can use the ______________________________theorem.

Note: Your calculator is used to evaluate a logarithm with base 10.

Example 5: Evaluate :

a.) 2log 14 b.) 5log 16

Applications of Logarithms

Transformations of Logarithm

State how each of the equations below transforms the graph of xxf alog)( = .

1. )1(log)( 2 += xxf

2. )5(log)( 3 −= xxf

3. 4( ) log (2 ) 3f x x= +

4. 5( ) 4 log ( ) 2f x x= −

5. )1(log)( 3 xxf −=

6. 1)4(log)( 2 +−= xxf

7. )62(log)( 2 += xxf

Example 1: Sketch each of the following and determine any x and y-intercepts.

a.) 3)1(log2 +−= xy

b.) )62(log)( 2 += xxf .

y

x

c.) )2(log)( 4 xxf −=

Example 2: The graph of 4logy x= can be transformed to produce the second graph as shown below. Write the equation of the transformed graphs.

a.) b.)

Example 3:

Laws of Logarithm

Product Law of Logarithms The logarithm of a product of numbers can be expressed as the _________ of the logarithms of the numbers.

( )log log loga a aMN M N= + Quotient Law of Logarithms The logarithm of a quotient of numbers can be expressed as the _____________ of the logarithms of the numbers.

log log loga a aM M NN

= −

Power Law of Logarithms The logarithm of a power of a number can be expressed as the ________________ times the logarithm of the number.

( )log logna aM n M=

Example 1: Write each expression in terms of individual logarithms of x, y, and z.

a.) 5log xyz

b.) 37log x

c.) 6 2

1logx

d.) 3

log xy z

Example 2: Use the laws of logarithms to simplify and evaluate each expression.

a.) 6 6 6log 8 log 9 log 2+ −

b.) 7log 7 7

c.) 2 2 212 log 12 log 6 log 273

− +

Example 3: Write each expression as a single logarithm in simplest form. State the restrictions on the variable. Restrictions:

a.) ( ) ( )22 2 24 log log 1 log 1x x x− − + +

b.) ( ) ( )25 5log 2 2 log 2 3x x x− − + −

c.) 2 77 7

5loglog log2

xx x+ −

Example 4: Use the approximations given and the laws of logarithms to perform the following calculation: log1.44 0.15836log1.2 0.07918log1.728 0.23754

≈≈≈

1.44

Example 5: The pH scale is used to measure the acidity or alkalinity of a solution. The pH of a solution is defined as logpH H + = − , where [H+] is the hydrogen ion concentration in moles per litre (mol/L). The pH scale ranges from 0 to 14. A neutral solution, such as pure water, has a pH of 7. Solutions with a pH of less than 7 are acidic and solutions with a pH of greater than 7 are basic or alkaline. The closer the pH is to 0, the more acidic the solution. Determine the hydrogen ion concentration when the pH is 5.5.

Logarithmic and Exponential Equations

Solving Logarithmic Equations Type 1 – Convert to Exponential Form Steps:

1.) Write as a ____________ logarithm. 2.) Convert to _________________ form. 3.) Solve for ______. 4.) Check for ___________________________ using the ____________________.

Example: Solve each of the following:

a.) ( )22log 3 4x + =

b.) ( )2 2log log 1 1x x+ + =

c.) ( ) ( )log 8 4 1 log 1x x+ = + +

d.) ( ) ( )5 5log 3 1 log 3 3x x+ + − =

Type 2 – Property of Equality Steps:

1.) Write as a _______________ logarithm on both sides (if necessary). 2.) Set ___________________ equal to each other (property of equality). Note: the

__________ must be the same to do this. 3.) Solve for _____. 4.) Check for ___________________________.

Example1: Solve for each of the following:

a.) ( )6 6log 2 1 log 11x − =

b.) ( )2 2 2log 2 log log 3x x− + =

c.) ( )5 5 5 5log 2 log 4 log 48 log 2t t+ − = −

Applications of Logarithms

Example 2: Palaeontologists can estimate the size of a dinosaur from incomplete skeletal remains. For a carnivorous dinosaur, the relationship between the length, s, in metres, of the skull and the body mass, m, in kilograms, can be expressed using the logarithmic equation: 3.6022log log 3.4444s m= − . Determine the body mass, to the nearest kilogram, of a dinosaur with a skull length of 0.78 m.

Example 3: If log 4 m= and log 3 n= , express log 48 in terms of m and n.

Solving Exponential Equations Using Logarithms It is not always possible to get a ____________________________ when solving

exponential equations. We can still solve these equations by using

____________________.

Steps:

1.) Log ____________ sides.

2.) Apply the _____________________ theorems.

3.) Get ____ to one side to solve.

Example 1: Solve for each of the following:

a.) 3 4x =

b.) ( )28 3 568x =

c.) ( ) 12 3 5x x+=

d.) 3 1 42 7x x− +=

Natural Logarithms

Natural logarithms are logarithms which are to the base of “e” where

e = ____________________ This can be found on your calculator. xxf elog)( =

xxf ln)( = OR

Example: Find the value of 2.3e .

2.3e = (using the calculator)

Example: Find )0051.0ln( . )0051.0ln( = Instead of using “log” to solve exponential equations that do not have the same base, you can use “ _____ ” (the symbol for natural log). You will get the same answer either way. Usually, ______ is used when solving equations that have an “ ___ “ because

Example: Solve: 3505.0 =− xe

Graphing: Graph: xy e= * Note that this is exponential.

y

x

Graph: lny x= * Note that this is logarithmic.

y

x

Note: xy e= is the ____________ of lny x= .

Applications of Exponential Functions Compound Interest Compounded Continuously where A = final amount P = initial amount r = rate (as a decimal) n = # of times it is compounded in one year t = time in years Remember: Annually once a year Semi-annually twice a year Quarterly four times a year Monthly twelve times a year Weekly fifty-two times a year Daily three hundred sixty-five times a year Examples 1. $100 is invested at a rate of 8% compounded semi-annually. How long will it take to

reach $1000?

2. How much money will there be if $22 000 is continuously invested at 10% over the course of 5 years?

3. What interest rate must be charged so that $2000 doubles in 10 years?

Often the following equation is used as a formula to represent exponential growth or decay. The variables and formulas may vary but they will be clearly given in the questions. Examples 1. There are 10 frogs in the swamp on October 31st. There are 60 frogs on November

15th. How many will there be on November 30th if they grow exponentially? * First determine the rate of growth. 2. A radioactive substance is decaying according to the formula 0.25ty A e−= • , where y

is the amount of material remaining after t years. Determine the half-life of the substance if the initial amount is 200 grams to the nearest hundredth of a year.