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f (x) ax

The inverse function of an Exponential functionsis a log function.

f 1(x) loga x

Domain:Range:Key Points:Asymptotes:

Graphing Logarithmic Functions

xaxalog

For each function below:a) List sequence of transformations and sketchb) State domain, range, and asymptotes c) Determine the inverse function.d) Sketch the inverse by reflecting graph of original function

1)3(log)( 1) 4 xxf

12)( 2) xxf

Section 4.5Properties of Logarithms

Condense and Expand Logarithmic Expressions.

Properties:1. Power Rule2. Product Rule3. Quotient Rule4. Condense an expression5. Expand an expression

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 lnlog

log lnaM M

Ma a

Change of Base Formula:

1. Power Rule

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

)ln( 1) 2x

)5(og 2) 45l

)ln( 3) x

22 )4(og 4) xl

2. Product Rule

“Expanding a logarithmic expression”Rewrite using the Product Rule.

))4(ln( 1) 32 xe

3)1)(4(og 2) xxl

3. Quotient Rule

“Expanding a logarithmic expression”Rewrite using the Quotient Rule.

24

16og 1)

xl

3

ln 2)3e

4. Expand the following expressions completely

1 23

2( 2)

2) ln1

x

x

32 25

log 1)xy

x

5. Condensing Logarithmic Expressions

Rewrite as a single log expression

32log2og 1) 44 l

)log(3-4xog 2) xl

1)ln(x4

1ln(x)2 3)

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

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

2xlog3

1-1)log(2x4log(x)

2

1 1)

More practice….