Download Logarithm Notes Part 1 Logarithm Rules Notes Part 1_2

You already know the parts of an Exponential Expression as a x = b a is the base €
x is the exponent b is the answer We define the parts of a Logarithmic Expression as log a b = x a is the base of the logarithm €
b is the argument of the logarithm x is the answer of the logarithm Logarithm Rules Already Covered: “A logarithm is an exponent” a x = b ⇔ log a b = x • A logarithm of 1 is equal to zero log a 1 = 0 €
• A logarithm of its own base value is always equal to 1 log a a = 1 €
€
•
Power Rule of Logarithms: A logarithm to a power can be expressed as a multiplication: log a b c
⇔ c log a b The reverse of this rule can “move” a leading term off a logarithm to be expressed as a power of that logarithm: €
c log a b ⇔ log a b c € Example: Evaluate: log 2 4 5 1) Remove the exponent off the logarithm: log 2 4 5
⇒ 5(log 2 4 ) € Evaluate log 2 4 2) log 2 4 = x
→ 2x = 4 → €
2 x = 2 2 → x = 2 € So, log 2 4 = 2 € 3) Substitute 2 into the expression in step 1: €
5(log 2 4 ) = 5(2) = 10 Example: €
Evaluate: 2log 4 16 1) Move the leading term to the argument as a exponent: €
2log 4 16
⇒ log 4
( 16 )
2
→
log 4 16 2) Evaluate log 4 16 : €
€
€
log 4 16 = x
→ 4 x = 16 → 4 x = 4 2
→ x = 2 Multiplication Rule of Logarithms: The sum of two logarithms can be combined into one logarithm if: The bases are the same The arguments are multiplied together as the new argument for the new logarithm. •
•
The reverse of this rule can be used to break a logarithm into a logarithm expression of two summed logarithms. log a b + log a c
log a bc
⇔ log a bc ⇔ log a b + log a c €
Example: €
Evaluate log 6 12 + log 6 3 1) Combine the two logarithms: log 6 12 + log 6 3 ⇒ log 6 (12)( 3) ⇒ log 6 36 €
2) Use the power rule property to quickly evaluate the logarithm: log 6 36 → log 6 (6 2 ) → log 6 6 2 → 2log 6 6 €
3) Any logarithm of itself is equal to 1. So, log6 6 = 1 € 4) Substitute the 1 into the expression in step 2: 2log 6 6 → 2(1) → 2 €
Division Rule of Logarithms: The difference of two logarithms can be combined into one logarithm if: The bases are the same The arguments are divided together as the new argument for the new logarithm. The numerator is the first argument and the denominator is the second argument. •
•
The reverse of this rule can be used to break a logarithm into a logarithm expression of two differenced logarithms. log a b − log a c
log a
b
c
b
⇔ log a c
⇔ log a b − log a c €
Example: €
Evaluate log 3 324 − log 3 4 1) Combine the two logarithms: €
log 3 324 − log 3 4 ⇒ log 3
324
4
⇒ log 3 81 2) Use the power rule property to quickly evaluate the logarithm: log 3 81 → log 3 ( 34 ) → log 3 34 → 4 log 3 3 €
3) Any logarithm of itself is equal to 1. So, log3 3 = 1 € 4) Substitute the 1 into the expression in step 2: 4 log 3 3 → 4 (1) → 4 €