Properties of Logarithms

1. Evaluate the expressions: log327 log2 ½ log255

2. Sketch the graph of f(x) = 4x and tell the domain, range, intercept, asymptote, and end behavior.

3. Sketch the graph of log4x and tell the domain, range, intercept, asymptote, and end behavior.

Warm Up

Friday’s homework answers

By the end of today, you will be able to…◦ Understand and use the properties of logarithms◦ Understand and use the change of base theorem

Today’s objectives

Properties of Logarithms They are just like the laws of exponents. Why does this make sense?

logb1 =

logbb =

logbbx =

Basic Properties of Logarithms

What about ln?

ln e =

Log10 =

logbxy = logbx + logby

What exponential rule does this remind you of?

Example: log354

Product Property

Quotient Property

yxy

xbbb logloglog

Example: log936

Power Property

xpx bp

b loglog

Examples: log4x9 5 ln e2 – ln e3

Evaluate this expression

Change of Base Formula

b

xx

a

ab log

loglog

This allows us to use our calculator to estimate log values that have bases other than 10 or e. (ln)

Our calculator only has the two common log buttons.

Evaluate each logarithm:

log35 log1/26

Example of Change of Base Formula

Rewrite the following expression using the product property:log20

Rewrite the following expression using the quotient property: log40

Rewrite the following expression using the power property: 3log10

Sticky Note Problem

They help us manipulate, evaluate, simplify, expand, and condense logarithmic expressions however we’d like to.

And ultimately, they help us solve equations with exponents and logarithms.

So why are these useful?

Asks us to: Make all coefficients exponents Rewrite the expression using only 1 log

Condensing Log Expressions

Condensing Log Expressions

Condense this expression:

4logx – 3log5x

Condense this expression:

6 ln(x – 4) + 3 lnx

Asks us to… Rewrite the expression using MORE THAN 1

log Make all exponents coefficients

Expanding Log Expressions

Expanding Log Expressions Expand this

expression: log12x5y-2

Expand this expression:

A look ahead…

The power property lets us get variables out of the exponent!

Remember our problem from Friday?300 = 200e.05t

4x+ 2 = 10

Complete the problems on the back of your guided notes sheet for today.

Working diligently earns you your class work stamp!

Whatever you do not finish in class is your homework.

p. 185 # 19 – 24, 29 – 32, 39 – 44, 49 – 58, p. 186 # 90 - 95

Practice Problems

Study for your quiz Wednesday (exponential word problems, graphs of exponential functions and logarithms)

Complete problems 21, 25, 37, 40 on the back of your notes from Thursday, as well as any other problems you did not finish today in class.

Homework

You have today, tomorrow, Thursday, and Friday afternoons to make test corrections if you would like to.

If you would like to know your current grade in the class, or have other questions, come see me after school!

Test Return