2.08 ExpLog - Logarithmic Manipulating Bases

University:
High School
Course:
High School Physics
Academic year:

2024
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8
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2.08 MANIPULATING PARAMETERS IN LOGARITHMIC RULES WITHOUT NUMBERS Standard Form 𝑓 𝑥 = 𝑎 log 𝑐 𝑏 𝑥 − ℎ By Product Law = 𝑎 log 𝑐 𝑏 + log 𝑐 𝑥 − ℎ + 𝑘 = 𝑎 log 𝑐 𝑏 + 𝑎 log 𝑐 𝑥 − ℎ + 𝑘 = 𝑎 log 𝑐 𝑥 − ℎ + 𝑎 log 𝑐 𝑏 + 𝑘 = 𝑎 log 𝑐 𝑥 − ℎ + 𝑘1 𝑙𝑜𝑔𝑐 𝑥 − ℎ = + 𝑘1 1 𝑎 𝑙𝑜𝑔𝑐 𝑥 − ℎ = + 𝑘1 1 log 𝑐 𝑐 𝑎 𝑙𝑜𝑔𝑐 𝑥 − ℎ = + 𝑘1 1 log 𝑐 𝑐 𝑎 = 𝑙𝑜𝑔 1 𝑥 − ℎ + 𝑘1 Expand Combine solvables 𝒌𝟏 = 𝒂 log 𝑐 𝒃 + 𝒌 Manipulate: × 𝒂 = ÷ 𝟏 𝒂 Manipulate: log 𝒄 𝒄 = 𝟏 By Base Law 𝑐𝑎 +𝑘 EXAMPLE 1 Standard Form Manipulate: × 𝒂 = ÷ 𝟏 𝒂 Manipulate: log 𝒄 𝒄 = 𝟏 By Base Law 𝑓 𝑥 = 2 log 9 𝑥 log 9 𝑥 = 1ൗ 2 log 9 𝑥 =1 2 log 9 9 log 9 𝑥 = log 9 91/2 log 9 𝑥 = log 9 3 = log 3 𝑥 CHANGING BASE: QUICK VERSION log 𝒄 𝒙 = □ log (𝒄 □) 𝒙 log 𝟐 𝒙 = log 𝟏𝟔 𝒙 = 𝟐 log 𝟒 𝒙 ൞ 𝟑 log 𝟖 𝒙 𝟒 log16 𝒙 (𝟒𝟏/𝟐 = 𝟐) (𝟖𝟏/𝟑 = 𝟐) (𝟏𝟔𝟏/𝟒 = 𝟐) 𝟏 𝟐 log 𝟒 𝒙 ቐ𝟏 𝟒 log 𝟐 𝒙 (𝟒𝟐 = 𝟏𝟔) (𝟐𝟒 = 𝟏𝟔) USE IN LOG SIMPLIFICATION! log 3 𝒙 + log 𝟗 𝒙 = TRY IT! log 4 𝒙 − log 𝟔𝟒 𝒙 = EXAMPLE 2 1 log 2 2𝑥 + 3 + 7 3 1 3 = − log 2 2 𝑥 + +7 3 2 Standard Form 𝑓 𝑥 =− By Product Law 1 3 =− log 2 2 + log 2 𝑥 + 3 2 Expand 1 1 3 = − log 2 2 − log 2 𝑥 + +7 3 3 2 Combine solvables 𝒌𝟏 = 𝒂 log 𝑐 𝒃 + 𝒌 Manipulate: × 𝒂 = ÷ 𝟏 𝒂 Manipulate: log 𝒄 𝒄 = 𝟏 𝟏 −𝟑 𝟐 = 𝟖 By Base Law +7 1 3 1 = − log 2 𝑥 + − log 2 2 + 7 3 2 3 1 3 20 = − log 2 𝑥 + + 3 2 3 3 log 2 𝑥 + 2 20 = + log 2 2−3 3 3 20 = 𝑙𝑜𝑔 1 𝑥 + + 2 3 8 HOMEWORK Worksheet: Changing Bases