2.13 ExpLog - Logarithmic Inequalities | Lecture Note

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2024
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2.13 SOLVING INEQUALITIES (EXP AND LOG) SOLVING EXPONENTIALS Ex 3: 𝟏 𝟓−𝟐𝒙 𝟑 ≥𝟒 Change of Base does not work Errors are often made switching forms with inequality symbol 𝟏 𝒍𝒐𝒈 𝟑 𝟓−𝟐𝒙 ≥ 𝒍𝒐𝒈 𝟒 𝟏 ≥ 𝒍𝒐𝒈 𝟒 𝟑 𝟏 𝟏 𝟓 𝒍𝒐𝒈 − 𝟐𝒙 𝒍𝒐𝒈 ≥ 𝒍𝒐𝒈 𝟒 𝟑 𝟑 𝟏 𝟏 −𝟐𝒙 𝒍𝒐𝒈 ≥ 𝒍𝒐𝒈 𝟒 − 𝟓 𝒍𝒐𝒈 𝟑 𝟑 𝟏 𝟏 𝒙 −𝟐 𝒍𝒐𝒈 ≥ 𝒍𝒐𝒈 𝟒 − 𝟓 𝒍𝒐𝒈 𝟑 𝟑 𝟓 − 𝟐𝒙 𝒍𝒐𝒈 Check if dividing by a negative! 𝒙≥ 𝒍𝒐𝒈 𝟒 − 𝟓 𝒍𝒐𝒈 𝟏 𝟑 𝟏 𝟑 𝒙 ≥ 𝟑. 𝟏𝟑𝟎𝟗𝟐𝟗𝟕𝟓𝟒 −𝟐 𝒍𝒐𝒈 3.1309, ∞ SOLVING EXPONENTIALS Ex 4: 𝟑𝟐𝒙−𝟏 > 𝟖𝒙+𝟐 Change of Base does not work 𝒍𝒐𝒈 𝟑𝟐𝒙−𝟏 > 𝒍𝒐𝒈 𝟖𝒙+𝟐 𝟐𝒙 − 𝟏 𝒍𝒐𝒈 𝟑 > 𝒙 + 𝟐 𝒍𝒐𝒈 𝟖 𝟐𝒙 𝒍𝒐𝒈 𝟑 − 𝒍𝒐𝒈 𝟑 > 𝒙 𝒍𝒐𝒈 𝟖 + 𝟐 𝒍𝒐𝒈 𝟖 𝟐𝒙 𝒍𝒐𝒈 𝟑 − 𝒙 𝒍𝒐𝒈 𝟖 > 𝟐 𝒍𝒐𝒈 𝟖 + 𝒍𝒐𝒈 𝟑 𝒙 𝟐 𝒍𝒐𝒈 𝟑 − 𝒍𝒐𝒈 𝟖 > 𝟐 𝒍𝒐𝒈 𝟖 + 𝒍𝒐𝒈 𝟑 Check if dividing by a negative 𝟐 𝒍𝒐𝒈 𝟖 + 𝒍𝒐𝒈 𝟑 𝒙> 𝟐 𝒍𝒐𝒈 𝟑 − 𝒍𝒐𝒈 𝟖 𝒙 > 𝟒𝟒. 𝟔𝟑𝟕𝟏𝟏𝟖𝟗𝟒 44.6371, 𝑃𝑂𝐼, ∞∞ 𝑦=0 SOLVING LOG INEQUALITIES Pay attention to domains and asymptote: If 𝒃 > 𝟎 𝒉, ∞ If 𝒃 < 𝟎 −∞, 𝒉 You will ALWAYS be given same base 𝒙=𝒉 1 𝑝𝑜𝑤𝑒𝑟: −2𝑥 + 1 = −2 𝑥 − 2 1 𝑑𝑜𝑚 𝑓 = −∞, 2 𝑓↘ EX 1: LOG5 1 − 2𝑥 < 3 Visualize 𝑦=3 𝑥= 1 2 Switch Forms: SIGN FOLLOWS POWER! 𝟏 − 𝟐𝒙 < 𝟓𝟑 𝟏 − 𝟐𝒙 < 𝟏𝟐𝟓 −𝟐𝒙 < 𝟏𝟐𝟒 𝒙 > −𝟔𝟐 𝟏 −𝟔𝟐, 𝟐 EX 2: −2 𝐿𝑁 1 𝑥 2 + 7 > −24 log 𝒆 𝟏 𝒙 + 𝟕 < 𝟏𝟐 𝟐 Visualize 𝑦 = −24 𝑥 = −14 Switch Forms: SIGN FOLLOWS POWER! 𝟏 𝒙 + 𝟕 < 𝒆𝟏𝟐 𝟐 𝒙 < 𝟐 𝒆𝟏𝟐 − 𝟕 𝒙 < 𝟑𝟐𝟓𝟒𝟗𝟓. 𝟓𝟖𝟐𝟖 −𝟏𝟒, 𝟑𝟐𝟓𝟒𝟗𝟓. 𝟓𝟖𝟐𝟖 5 𝑝𝑜𝑤𝑒𝑟: 3𝑥 − 5 = 3 𝑥 − 3 EX 3: LN 𝑥 ≥ LN 3𝑥 − 5 Visualize 𝑥=0 Same Base: DROP THEM 5 𝑥= 3 𝒙 ≥ 𝟑𝒙 − 𝟓 −𝟐𝒙 ≥ −𝟓 𝟓 𝒙≤ 𝟐 𝟓 𝟓 , 𝟑 𝟐 TRY IT log 𝟒 𝟑𝒙 − 𝟗 ≤ log 4 𝟐𝒙 PRACTICE Visions 1 P199: 6efgh