Essential Question: What is the relationship between a logarithm and an exponent?

You’ve ran across a multitude of inverses in mathematics so far...◦ Additive Inverses: 3 & -3◦ Multiplicative Inverses: 2 & ½ ◦ Inverse of powers: x4 & or x¼

◦ But what do you do when the exponent is unknown? For example, how would you solve 3x = 28, other than guess & check?

◦ Welcome to logs…

4 x

Logs◦ There are three types of commonly used logs

Common logarithms (base 10) Natural logarithms (base e) Binary logarithms (base 2)

◦ We’re only going to concentrate on the first two types of logarithms, the 3rd is used primarily in computer science.

◦ Want to take a guess as to why I used the words “base” above?

The logarithm to the base b of a positive number y is defined as follows:◦ If y = bx, then logby = x

◦ All logs can be thought of as a way to solve for an unknown exponent logbase answer = exponent

log10

10 2x =

x

10 2x =

Example: Write “25 = 52” in logarithmic form◦ Remember: logbase answer = exponent

◦ log5 25 = 2

Your turn: Write the following exponential equations in logarithmic form.◦ 729 = 36

◦ (1/2)3 = 1/8

◦ 100 = 1

log3 729 = 6

log1/2 1/8 = 3

log10 1 = 0

Example: Write “log8 16 = x” in exponential form◦ Remember: logbase answer = exponent◦ 8x = 16

Your turn: Write the following exponential equations in logarithmic form.◦ log64 1/32 = x

◦ log9 27 = x

◦ log10 100 = x

64x = 1/32

9x = 27

10x = 100

Assignment◦ Page 450◦ Problems 7 – 25 & 53 – 61 (odd problems)

For questions 15 – 25, pretend they “= x” We’ll deal with how to solve them Tuesday

Essential Question: What is the relationship between a logarithm and an exponent?

Common logarithms◦ Scientific/graphing calculators have the common

and natural logarithmic tables built in.◦ On our calculators, the “log” button is next to

the carat (^) key.◦ To find log10 29, simply type “log 29”, and you

will be returned the answer 1.4624. That means, 101.4624 = 29

◦ Though the calculator will give you logs to a bunch of places, round your answers to 4 decimal places

Solving Logarithmic Equations (w/o calculator)◦ log216 = x

Can be rewritten as 2x = 16. Because 24 = 16, x = 4

◦ log5(-25) = x Rewritten as 5x = -25, which isn’t possible. Undefined

◦ log5x = 3 Can be rewritten as 53 = x, so x = 125

Change-of-Base Formula

◦

◦ Find log89

loglog

logb

vv

b

8

log 9log 9 1.0566

log 8

Example: Solve for xlog8 16 = x

Your turn: Solve for x◦ log64 1/32 = x

◦ log9 27 = x

◦ log10 100 = x

-0.8333

1.5

2

8

log16log 16 1.3333

log 8

Assignment◦ Page 450◦ Solve problems 41 – 47, odds◦ Solve problems 15 – 25, odds

◦ Page 464◦ Solve problems 25 – 31, odds