4.4 Evaluate Logarithms and Graph Logarithmic Functions

Part 2

Definition

• Logarithms are the "opposite" of exponentials,

• Logs "undo" exponentials.

• Logs are the inverses of exponentials.

Writing Logarithms

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____________________________________________cab logYou read it: Log base “b” of “a” equals “c”

‘log’ is the operation b is the base a is the object of the log c is what you get when you evaluate the log

Exponential Form

log x yb =

x yb =Logarithmic Form

5 1x 416 x13 x

Evaluating logarithms now you try some!

• Log 4 16 =

• Log 5 1 =

• Log 16 4 =

• Log 3 (-1) =(Think of the graph of y = 3x)

20

½ (because 161/2 = 4) undefined

4 16x

You should learn the following general

forms!!!

•Log a 1 = 0 because a0 = 1

•Log a a = 1 because a1 = a

•Log a ax = x because ax = ax

Common logarithms

•log x = log 10 x

•Understood base 10 if nothing is there.

Common Logs and Natural Logs with a

calculator

log10 button

lne button

Finding Inverses

• Find the inverse of:

•y = log3x

• By definition of logarithm, the inverse is

y=3x

• OR write it in exponential form and switch the x & y!

3y = x 3x = y

Example 1:

• Write 53 = 125 in logarithmic form.

• Write log381 = 4 in exponential form.

Try This: Complete the table.

Exponential Form

25 = 32

3-2 = 1/9

Logarithmic Form

log101000 = 3

Log164 = 1/2

#1 #2 #3 #4

Lets look at their graphs

y = x

10xy

10log y x

10logy x

To Evaluate Logs without a Calculator

• Change the log to an exponential.

1. log2 32 = x 2. log4 2 = x

Solve for x.

1. log2 64 = x 2. logx 343 = 3

Change the log to an exponential.

Evaluate without a calculator:

1. log 2 8 = x

2. log 2 1 = x

3. Find the value of k : k = log 9 3

4. Find the value of k : ½ = log k 9

5. Find the value of k : 3 = log 7 k

Change the log to an exponential.

Common Logarithms

• Logarithms with base ______ are called common logarithms.

• Sometimes the base is assumed and not written.

• Thus, if you see a log written without a base, you assume the base is _______.

• The log button the calculator uses base _____.

10

10

10

Use your calculator to evaluate:

1. log 51

2. log 4

3. log 0.215

1.71

0.6

– 0.67

Which means 1.7110 51

Do You Know What X is?

4. Solve for x: 10x = 728

5. Solve for x:

Change the exponential to a log. Then use calculator.

1085

110 x

Remember e ?

2.718e

Natural Logarithm

• A natural logarithm is a logarithm with base e, denoted by ln.

• A natural logarithm is the inverse of an exponential function with base e.

xxe lnlog

2 7.389e

Exponential Form Logarithmic Form

ln 7.389 2

Lets look at their graphs

y = x

xy e lny x

ln y x

Writeasexponent or log.

1. 4

2. ln 56.3 4.03

xe

Evaluate f(x)=ln x to the nearest thousandth for each value of x below:

1.52

1.42.3 xxx

0.693 – 0.693 ? (see graph)

lny x

13. Find the inverse of y = ln(x+1)

14. Find the inverse of y = 5x .

y = ex - 1

y = log5x

Homework

Book

Pg. 147 16 - 24 allPg. 148 13 – 21 all