PC

Mr. Sutton presents…

5 Exponential and Logarithmic Functions 5.10 Laws of Logarithms

Standard: F.LE.4

Objective


To simplify expressions using the laws of logarithms (log of a product/difference, log of a power)

Key Terms and Properties: Log Rules One Way

log(10 ⋅100)

= log(1000) = 3

Another Way

= log10 + log100 = 1 + 2 = 3

Observation: taking the log of a product of two numbers is the same as adding the logs of those numbers. Log turns multiplication into addition!

RULE: log xy = log x + log y One Way

 100,000  log   = log(1000) = 3  100 

Another Way

= log100,000 − log100 = 5 − 2 = 3

Observation: taking the log of a quotient of two numbers is the same as subtracting the logs of those numbers. Log turns division into subtraction!

RULE: log x / y = log x − log y

Key Terms and Properties: The “Pop Out” Exponent Rule One Way log(1003 )

Another Way = log(100 ⋅100 ⋅ 100) = log100 + log100 + log100 = 2+2+2 = 6

= log(1,000,000) = 6

Yet Another Way log(1003 )

= 3log100 = 3 ⋅ 2 = 6

Observation: if we take the log of something with an exponent, we can just bring the exponent out in front. Logs turn exponents into multiplication!

n

RULE: log x = n log x

Process: Breaking Expressions Down Into Logs








1.) If the whole expression is being raised to an exponent (see E2), take that out first and distribute it 2.) Apply the addition/subtraction rule 3.) Take each term’s exponent outside the log

Write in terms of log x and log y

x6 log 3 y = log x 6 − log y 3 E1

= 6log x − 3log y log 5 xy 1 = log xy 5

E2

1 1 = log x + log y 5 5

Process: Writing An Expression as a Single Log








1.) Apply numbers outside logs as exponents

Write as a rational number or single log

9log x + log y − 2log z

= log x9 + log y − log z 2

2.) Turn addition into multiplication E3 and subtraction into division

x9 y = log 2 z log 2000 + 4log 5 − 3log 5 = log 2000 + log 54 − log 53

3.) Simplify, if possible

= log(2000 ⋅ 625 ÷ 125)

E4

= log10000 = 4

Process: Practice Writing An Expression as a Single Log Write as a rational number or single log

log 6 24 + 2log 6 3

ln 54 − 3ln 3

= log 6 24 + log 6 3

= ln 54 − ln 33

= log 6 (24 ⋅ 9)

54 = ln 27

2

= log 6 216

= ln 2

=3

E5

E6

Process: More Than One Thing in the Exponent








If log or ln has a coefficient, make it an exponent first

Simplify

E7

Recall that 10log(x) = x and eln(x) = x. Split up the sum and differences of exponents into multiplication and division of like bases

10

e

1 − ln x 6

4 log 2

=e

= 10

ln x −1/ 6

log 24

= 16

1 =6 x

E8

e14+ ln 8 = e14 ⋅ eln 8 = 8e14 E9