M3U7D3 Warm Up

x = 2

Solve each equation. 1. 8 = x3

2. x½=43. 27 = 3x

4. 46 = 43x

Graph the following:5. y = 2x2

x = 16x = 3

x = 2

Homework Check:

Document Camera

U7D3 Log Properties

OBJ: For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

F-LE.4

First let’s summarize the First let’s summarize the properties we discovered properties we discovered

observing the classwork we observing the classwork we completed and checked in completed and checked in

lesson 7lesson 7

NOTICE!!!NOTICE!!!

2200 = 1 = 1 LogLog22 1 = 0 1 = 0

2211 = 2 = 2 LogLog22 2 = 1 2 = 1

2222 = 4 = 4 LogLog22 4 = 2 4 = 2

2233 = 8 = 8 LogLog22 8 = 3 8 = 3

2244 = 16 = 16 LogLog22 16 = 4 16 = 4

2255 = 32 = 32 LogLog22 32 = 5 32 = 5

Properties of Logarithms

There are four basic properties of logarithms that we will be working with. For every case, the base of the

logarithm can not be equal to 1

and the values must all be positive (no negatives in logs)

Product Rule

loglogbbMN = logMN = logbbM + logM + logbbNN

Ex: logEx: logbbxy = logxy = logbbx + logx + logbby y

Ex: log6 = log 2 + log 3Ex: log6 = log 2 + log 3

Ex: logEx: log339b = log9b = log339 + log9 + log33bb

Quotient Rule

yxy

x555 logloglog

P

MN2log

NMN

Mbbb logloglog

5loglog5

log 222 aa

PNM 222 logloglog

Power Rule

BB 52

5 log2log

437log ba

MxM bx

b loglog

5log5log 22 xx ba 77 log4log3

Change of Base Formula

• Example loglog558 =8 =

• This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

log

log.

8

512900

b

logMlog M

logb

These next two problems tend to be some of the trickiest to evaluate.

Actually, they are merely identities and

the use of our simple rule will show this.

Example 1: Evaluate: log7 72

Solution:

Now take it out of the logarithmic form and write it in exponential form.

log7 72 y First, we write the problem with a variable.

7y 72

y 2

Example 2: Evaluate: 4 log4 16

Solution:

4 log4 16 y First, we write the problem with a variable.

log4 y log4 16 Now take it out of the exponential form and write it in logarithmic form.

y 16

Ask your teacher about the last twoexamples.

They may show you a nice shortcut.

If Loga ab = y then y = b AND…

If aLoga b = y then y = b

Finally, we want to take a look at the Property of Equality for Logarithmic Functions.

Finally, we want to take a look at the Property of Equality for Logarithmic Functions.

Suppose b 0 and b 1.

Then logb x1 logb x2 if and only if x1 x2

Basically, with logarithmic functions,if the bases match on both sides of the equal sign , then simply set the arguments equal.

Basically, with logarithmic functions,if the bases match on both sides of the equal sign , then simply set the arguments equal.

Example 3:

Solve: log3 (4x 10) log3 (x 1)

Solution:Since the bases are both ‘3’ we simply set the arguments equal.

4x 10 x 13x 10 1

3x 9x 3

Example 4:Solve: log8 (x2 14) log8 (5x)

Solution:

Since the bases are both ‘8’ we simply set the arguments equal.x2 14 5xx2 5x 14 0(x 7)(x 2) 0

Factor

(x 7) 0 or (x 2) 0x 7 or x 2

Solve: log8 (x2 14) log8 (5x)

Solution:x 7 or x 2

But we’re not finished…

Example 4 continued…Solve: log8 (x2 14) log8 (5x)

x 7 or x 2

It appears that we have 2 solutions here.If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive

bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution.

Let’s end this lesson by taking a closer look at this.

Our final concern then is to determine why logarithms like the one below are undefined.

Our final concern then is to determine why logarithms like the one below are undefined.

Can anyone give us an explanation ?

Can anyone give us an explanation ?

2log ( 8)

One easy explanation is to simply rewrite this logarithm in exponential form. We’ll then see why a negative value is not permitted.

First, we write the problem with a variable.

2y 8 Now take it out of the logarithmic form and write it in exponential form.

What power of 2 would gives us -8 ?

23 8 and 2 3 1

8Hence expressions of this type are undefined.

2log ( 8) undefined WHY?

2log ( 8) y

Example 5:

Natural Logs

• A logarithm to the base e (2.71828…).

• Written ln (pronounced ell-n)

• Can be accessed on your calculator using LN or 2nd LN to get to ex.

Natural Logs and “e”

Start by graphing y=ex The function y=ex has an inverse called the Natural

Logarithmic Function.

Y=ln x

What do you notice about the

graphs of y=ex and y=ln x?

y=ex and y=ln x are inverses of each other!

We can use the natural log to “undo” the function y= ex (and vice versa).

All the rules still apply• You can use your

product, power and quotient rules for natural logs just like you do for regular logs

4ln2ln5

8ln4

32ln

4

2ln

5

Example 6:

yx lnln3 )ln( 3 yx

Example 7:

yx 3ln

OR

That concludes our introduction to logarithms. In the lessons to follow we will learn some important properties of logarithms.

One of these properties will giveus a very important tool whichwe need to solve exponential equations. Until then let’s practice with the basic themes of this lesson.

That concludes our introduction to logarithms. In the lessons to follow we will learn some important properties of logarithms.

One of these properties will giveus a very important tool whichwe need to solve exponential equations. Until then let’s practice with the basic themes of this lesson.

Sum of Properties of General and Sum of Properties of General and Natural LogarithmsNatural Logarithms

General Properties Natural Logarithms

1. logb 1 = 0 1. ln 1 = 0

2. logb b = 1 2. ln e = 1

3. logb bx = 0 3. ln ex = x4. b logb x = x 4. e ln x = x

REMEMBER Common Logarithms are logs base 10.

Classwork

M3U7D3 Investigating the Properties of Logarithms part I

Homework

M3U7D3 packet pages 3&4 Properties of Logarithms

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