5.10 Laws of Logarithms. Standard: F.LE.4. Page 2. Objective. â«. To simplify expressions using the laws of logarithms
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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
