Switch x and y.

3yx

3-1= 1/3

30= 1

31= 3

32= 9

b > 0 and b = 1

V.A. at x = 0

Common Point at ( 1, 0 )

Domain: Range: , ,0

b > 1: increase; 0 < b < 1: decrease

b > 0 and b = 1

H.A. at y = 0

Common Point at ( 0, 1 )

Domain: Range: , ,0

b > 1: increase; 0 < b < 1: decrease

SWITCH X & Y CONCEPTS!

Negative, flip over the x-axis. 0 < a < 1, Vertical Shrink and a > 1, Vertical Stretch.

Negative, flip over y-axis. 0 < b < 1, Horizontal Stretch and b > 1, Horizontal Shrink.

Solve for x. This is the Horizontal shift left or right.

This is the Vertical shift up or down.

Translate

Graph

The negative will flip the graph over the x-axis.

x = 0

xxf 3log

xxf 3log

Translate

Graph

The minus 2 is inside the function and solve for x. x = +2, shift to the right 2 units.

x = 0

xxf 3log

2log3 xxf

x = 2

Translate

Graph

The minus 5 is outside the function and shift down 5 units.

xxf 3log

5log3 xxf

x = 0

Translate

Graph

The negative on the x will flip the graph over the y-axis and solve 1 – x = 0 to determine how we shift horizontally. x = 1

x = 0

xxf 3log

xxf 1log3

x = 1

Translate

Graph

The plus 2 is inside the function and solve for x. x = -2, shift to the left 2 units. The minus 1 will shift down 1 unit.

x = 0

xxf 3log

12log3 xxf

x = -2

Translate

Graph

The negative on the x will flip over the y-axis. The negative in front of the log will flip the graph over the x-axis.

x = 0

xxf 3log

xxf 3log

42 x 3125log5

Base

Exponent

38

1log2

11logm

4972 324 5.2 772

WE MUST MEMORIZE THIS CONVERSION RULE!

Base e is the value 2.718281828…...

15log3 10 xyReplace f(x) with y.

Notice that there is no base number listed. Put in a 10.

154 2 xySwitch x and y. 15log3 10 yx

Solve for y. 5log31 10 yx

5log3

110

y

x

Convert to exponential.510 3

1

yx

yx

510 3

1 510 3

11

x

xf

154 2 yx

2541 yx

254

1 yx

24

1log5

yx

Convert to logarithm.

yx

24

1log5

24

1log5

1

xxf

Isolate the log function or exponential function.

The log function is isolated.Covert to exponential.

7432 x

749 x

x416

x4

642 x

8x

8xHas to be +8 because the base must be positive.

The exponential function is NOT isolated.Divide by 3.

102 xeCovert to logarithmic.

x210ln

xe 210log

Should be written with ln.

x

2

10ln

The base of 2 on the log and inside the ( )’s cancel out.

3 112 2log

3 11

The base of e and the ln in the exponent cancel out.

13lne

13

The 2 log base 2 can be condensed into one log.

16log3

48log

3log48log

22

22

42log 42

16 = 2*2*2*2

Set the exponent = x.

xe

9log 2

Use the base change formula for both logs.

8log

9log

3log

8log

The 2 log base 6 can be condensed into one log.

36log49log

4log9log

66

66

26log 26

36 = 6*6

Convert to exponential.

92 x

e

92 xeConvert to natural log and solve for x.

2

1

9ln9ln2

1

9ln2

x

x

33ln9ln ee

3log

9log

23log

9log2

3

3

Place the value of x back as an exponent on e.

The goal here is to factor the argument of the inside the log function to a 2, 3, or 5.

322log b

3 4

2 2

3log2log2log bbb 0986.16931.06931.0

4848.2

Convert the decimal to a fraction.

10

27logb

3 3 3

5 2

52

3log

3

bProduct RuleQuotient Rule

5log2log3log 3bbb

Power Rule 5log2log3log3 bbb

6094.16931.00986.13 9933.0

2 3 5 b

bbbbb log5log3log2log

16094.10986.16931.0

4011.4

2

1

6logb 6log2

1b

Power Rule

32log2

1 b 3log2log

2

1bb

0986.16931.02

1 89585.07917.1

2

1

Change-of-Base formula is

bM

b

MMb ln

ln

log

loglog

13log

39log39log13

Conversion Check

21log

5log5log21

11log

8log8log

11

13ln

39ln39log13

21ln

5ln5log21

11ln

8ln8log

11

We will use the Product, Quotient, and Power Rule to expand. Simplify, if possible.

Product Rule

yx 22

22 loglog8log

Power Rule

yx 222 loglog28log

Breakdown the 8 = 23

yx 223

2 loglog22log

yx 22 loglog23

Notice that there is a log for every factor. This is true whether the factors are top or bottom. Remember that the factors from the bottom are always minus the log.

Count the factors.1 2

3 4

A log for each.

33333 1log9loglog4log xy

Plus logs from the top. Minus logs from the bottom.

Simplify individually with Power Rule.

333

2

1

33 1log9loglog4log

xy

1log39loglog2

14log 3333 xy

One more.

1log33loglog2

14log 3

2333 xy

1log32log2

14log 333 xy

We will use the Product, Quotient, and Power Rule to condense. Simplify, if possible.

We must have a log in every term!The 2 is the answer from a simplified log. This means that is was the exponent on the base inside of a log5(5).

11log2

1log35loglog4 5555 yx

2

11log2

1log35loglog4 55

255 yx

Move all coefficients back inside as powers.

2

1

53

52

54

5 11loglog5loglog yx

Simplify the insides, if possible.

11loglog25loglog 53

554

5 yx

Optional.

3

4

5 25

11log

y

x

Plus logs to the top. Minus logs to the bottom.

We have a log in every term!

1

1log

2

3

3 x

x

top bottom

Simplify the inside…factor.

11

11log

2

3 xx

xxx

Perfect Cube

Diff. of Squares

1

1log

2

3 x

xx

One-to-one Property Conversion Rule

1

33

213

x

x

x

1332 x

We have a single log, convert to exponential.

139 x

x

x

3

10

310

Solve

Not yet One-to-one Property.Power Rule to move coefficients. 3

52

5 4loglog x

Now, One-to-one Property.Cancel logs.

32 4x

642 x

642 x

8xThe Domain must be > 0…no negative 8 as a solution.

8x

Condense to one log on the left side. Use the Power and Product Rule.

54log1log3 22 x

54log1log 23

2 x

514log 32 x

Now that there is one log, convert to exponential

35 142 x 32

81 3 x

21x

3x

3 3

2 2log 7 log 8 1x x Solve

24 4log 9 log 3 3x x

Condense to one log on the left side. Use the Product Rule.

187log2 xx

Now that there is one log, convert to exponential

8721 xx

base exponent

FOIL

56152 2 xxSet = 0.

054152 xxFactor

9,6

096

x

xx

Now check answers…-9 doesn’t work. -9 creates negative values inside the log.

Condense to one log on the left side. Use the Quotient Rule.

3

3

9log

2

4

x

x

Factor and simplify the inside.

3

3

33log4

x

xx

33log4 xNow that there is one log, convert to exponential

343 x 64

67x

Now we isolate the exponential expression.

153 x Convert to a log.

x15log3

Our answer for x is considered an exact value. You may be asked to convert to a rounded decimal answer.Base Change Formula.

This is factorable. Notice the first power is double that of the second power.

122 uu

34 uu 03242 xx

Set = 0 and solve for x.

042 x 032 x

42 x 32 x

Convert to logs for both. x 4log2 x3log2

No negatives in log functions or exponentials can’t = negatives

This is NOT factorable. Notice the first power is NOT double that of the second power.

FORCE IT! AARRRRGGGGG!Use your exponential rules. When you multiply like bases, we add the exponents. So manipulate the middle term.

xxx 33333 11

043332 xx 14432 uuuu

01343 xxSet = 0 and solve for x.

043 x 013 x

43 x 13 xNot Possible

Convert to a log.

x1log3

0x

UGLY! Bases are different.Authors way.

The author wants you to take either the common log or natural log of both sides.

xx 211 5ln2ln Power Rule

5ln212ln1 xx Dist. Prop.

5ln25ln2ln2ln

5ln25ln12ln12ln

xx

xx

Move all the terms with an x to the left side. 5ln2 x 5ln2 x

5ln2ln5ln22ln xx 2ln 2ln

2ln5ln5ln22ln xx

Non- x terms to the right side.

Factor out x as GCF.

2ln5ln5ln22ln x Isolate x.

5ln22ln 5ln22ln

5ln22ln

2ln5ln

x Power Rule

25ln2ln

2ln5ln

x Product &Quotient

Rule

252ln25

ln

x

50ln

5.2lnx

Base Change

5.2log50x

Mr. Fitz’s way.Pick a base and convert it to a log of that base.

xx 212log 15

I will go base 5.

Power Rule

xx 212log1 5 Dist. Prop. xx 212log2log 55 Move all the

terms with an x to the left side. 12log22log 55 xx

x2x2

Non- x terms to the right side.

2log5 2log5 2log122log 55 xx

Factor out x as GCF. 2log122log 55 x

Isolate x. 22log5 22log5

22log

2log1

5

5

x

2

55

51

5

5log2log

2log5log

x

25

5

52log25

log

x

5.2log50log

5.2log50

5

5 x