Warm Up:Simplify the Expression. Then State

briefly how to simplify the exponents.

Warm Up:Simplify the Expression. Then State

briefly how to simplify the exponents.x228

2

10

e

e

3910b

1.

2.

3.

4-5 Properties of Logarithms4-5 Properties of Logarithms

Ex 1: Condense the Expression

x22 log8log a.

b.

xx 88 222

Remember that Multiplication and Addition go hand-in-hand.

x8log2We must have the same base to condense!!!2ln10ln ln is in base e.

82

10

ee

e

Division and Subtraction go hand-in-hand.

2

10ln

5ln

Ex 3: Condense the ExpressionEx 3: Condense the Expression

xy4log3 27339 1010 bb

Here, the distributive law can be used.

34log xy33

4log yx

916log z

Let’s Put it all together…Let’s Put it all together…

Ex 4&5: Condense the expression.

zlog94log2 92 log4log z

924log z

xy 777 log23loglog )log23(loglog 777 xy )log3(loglog 2

777 xy )3(loglog 2

77 xy

27 3log

x

y

We can Also go Backwards!We can Also go Backwards!

45log We can also “un-distribute” the exponent

5log4

Ex6: Expand the expression.

We can Also go Backwards!!We can Also go Backwards!!

Ex 6&7: Expand the expression.

52 4log k

522 log4log k

k22

2 log52log

2

2lnv

x

vx ln22ln

vx ln2ln2ln k2log52

2ln2ln vx

Ex 7: Use log 3 ≈ .477 and log 2 ≈ .301 to approximate the value of

the expression

Ex 7: Use log 3 ≈ .477 and log 2 ≈ .301 to approximate the value of

the expression

12log

34log 322log

3log2log2log

The Change of Base FormulaThe Change of Base Formula

Let u, b, and c be positive numbers with b≠1 and c≠1. Then:

c

uu

b

bc log

loglog

More specifically:

c

uuc log

loglog

c

uuc ln

lnln

Ex 8: Use the change of base formula to evaluate the

expression

5log99log

5log

954.

699. 732.

Ex 1: Use properties of logarithms to evaluate the expression

Ex 1: Use properties of logarithms to evaluate the expression

279log3

323 33log

53 3log = ?

?3 53 5?

)000,10log(a. b.

)10log( 4

4