Properties of Logarithms

log 1a loga a log ra a loga ra

Product Rule: log ( ) log loga a aMN M N

Quotient Rule: log log loga a aM M NN

Power Rule: log logra aM r M

log lnloglog lnaM MMa a

Change of Base Formula:

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

f (x) ax

The inverse function of an Exponential functionsis a log function.

f 1(x) loga xDomain:Range:Key Points:Asymptotes:

Graphing Logarithmic Functions

xaxalog

Section 4.5Properties of Logarithms

Condense and Expand Logarithmic Expressions.

Rewrite expression to get same base on each side of equal sign.

where u and v are expressions in x

If au av , then uv

8221 2) x

xx 23 33)1

Type 1. Solving Exponential Equations

321641 3) x

Exponential Equations with base eTreat as a number.

3

2 1 2

eee xx

vuee v then ,ue

xx

xx ee

e

1

56 )(e 4)2 ee xx

Rewrite these expressions to have a single base e on both sides of the equation

Type 2 Solving: Log = Log

If then u = v vu aa loglog

)(log2log 5) 233 xx

When solving log functions, we must check that a solution lies in the domain!

)64ln(13ln( 6) x)x

Type 3. Solving: Log ( ) = Constant• Isolate and rewrite as exponential

3)12(og 7) 2 xl

2)7(log6-32 8) 23 x

4)6(log4 9) 27 xx

)1ln( 10-8007 10) t

Type 4: Exponential = ConstantIsolate exponential part and rewrite as log

210 )11 3 m

55 12) 1 xe

9)21(4 :one Try this 1 x

1. Power Rule“Expanding a logarithmic expression”Rewrite using the power rule.

15 )3(og 2) xl

)ln( 1) x

2. Product Rule“Expanding a logarithmic expression”

Rewrite using the Product Rule.

))4(ln( 2) 32 xe

3)1)(4(og 3) xxl

)3(og 1) 45 xl

3. Quotient Rule“Expanding a logarithmic expression”

Rewrite using the Quotient Rule.

2416og 1)x

l

4. Expand the following expressions completely

1 23

2( 2)2) ln

1xx

xx

25log 1) 2

5. Condensing Logarithmic Expressions

Rewrite as a single log expression

32log2og 1) 44 l

)log(3-4xog 2) xl

1)ln(x41ln(x)2 3)

Coefficients of logarithms must be 1 before you can condense them.

233 3 32) 15log log 9 log 9x x

2xlog31-1)log(2x4log(x)

21 1)

More practice….

7. Change-of-Base Formulalog lnloglog lnaM MMa a

Example.Find an approximation for )5(log2