2.12 ExpLog - Logarithmic Equations | Lecture Note

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2024
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2.12 SOLVING LOGARITHMIC EQUATIONS FINDING 𝑦 GIVEN 𝑥 Ex 1: Find 𝒇 𝟎 𝒇 𝒙 = 𝟏𝟐 log 𝟓 𝟑 − 𝒙 + 𝟖 𝒇 𝟎 = 𝟏𝟐 log 𝟓 𝟑 − 𝟎 + 𝟖 = 𝟏𝟐 log 𝟓 𝟑 + 𝟖 ≈ 𝟏𝟐 𝟎. 𝟔𝟖𝟐𝟔 + 𝟖 ≈ 𝟏𝟔. 𝟏𝟗𝟏𝟐𝟕 FINDING 𝑦 GIVEN 𝑥 Ex 3: Find 𝒇 𝟏𝟔 𝒇 𝒙 = −𝟑 log 𝟏 −𝒙 + 𝟓 𝟒 𝒇 𝟏𝟔 = −𝟑 log 𝟏 −𝟏𝟔 + 𝟓 𝟒 POWER ≠ NEGATIVE 𝒙 = 1𝟔 𝑫𝑵𝑬 FINDING 𝑥 GIVEN 𝑦 Ex 4: Find the Zero 𝒇 𝒙 = 𝟓 log 𝟏 𝟐(𝒙 − 𝟏 ) + 𝟏𝟎 𝟏𝟎 𝟎 = 𝟓 log 𝟏 𝟐(𝒙 − 𝟏 ) + 𝟏𝟎 𝟏𝟎 −𝟐 = log 𝟏 𝟐(𝒙 − 𝟏 ) 𝟏𝟎 𝟏 𝟐 𝒙−𝟏 = 𝟏𝟎 −𝟐 𝟐 𝒙 − 𝟏 = 𝟏𝟎𝟎 𝒙 − 𝟏 = 𝟓𝟎 𝒙 = 𝟓𝟏 TRY IT Solve for 𝒇 𝒙 = 𝟖 𝒇 𝒙 = log 𝟐 𝒙 − 𝟑 + 𝟒 𝑥 = 19 FINDING 𝑥 GIVEN 𝑦 Ex 6: 𝒇 𝒙 = log 𝟔 𝒙 + 𝟑 𝒈 𝒙 = − log 𝟔 𝒙 − 𝟐 + 𝟏 Find 𝒇 𝒙 = 𝒈(𝒙) log 𝟔 𝒙 + 𝟑 = − log 𝟔 𝒙 − 𝟐 + 𝟏 log 𝟔 𝒙 + 𝟑 + log 𝟔 𝒙 − 𝟐 = 𝟏 Restrictions: log 𝟔 𝒙 + 𝟑 𝒙 − 𝟐 = 𝟏 𝟔𝟏 = 𝒙 + 𝟑 𝒙 − 𝟐 𝟔 = 𝒙𝟐 + 𝒙 − 𝟔 𝟎 = 𝒙𝟐 + 𝒙 − 𝟏𝟐 𝟎= 𝒙+𝟒 𝒙−𝟑 𝒙 = −𝟒, 𝟑 Power > 0 (1) 𝑥 + 3 > 0 𝑥 > −3 (2) 𝑥 − 2 > 0 𝑥>2 Must satisfy both 𝒙=𝟑 TRY IT 2 Solve for 𝒇 𝒙 = 𝒈 𝒙 𝟏 𝒇 𝒙 = log 𝟖 𝟐𝒙 − 𝟕 − 𝟏𝟏 𝟑 𝑥=4 𝒈 𝒙 = 𝟏 log 𝟖 𝟓 − 𝒙 − 𝟏𝟏 𝟑 HOMEWORK Visions 1 P199 5deg, 9abc 4.12B SOLVING WITH LOGARITHMS SOLVING EXPS USING LOGS METHOD 1 METHOD 2 −𝟑𝟎 𝟎. 𝟗 𝒙 𝟐 + 𝟏𝟎 = −𝟓 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝐵𝑎𝑠𝑒 𝟎. 𝟗 𝒙 𝟐 + 𝟏𝟎 = −𝟓 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝐵𝑎𝑠𝑒 𝟏 = 𝟐 𝑆𝑤𝑖𝑡𝑐ℎ 𝐹𝑜𝑟𝑚𝑠 𝒙 𝟏 = log 𝟎.𝟗 𝟐 𝟐 𝑆𝑜𝑙𝑣𝑒 𝐿𝑜𝑔 𝒙 ≈ 𝟔. 𝟓𝟕𝟖𝟖 𝟐 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝑥 −𝟑𝟎 𝟎. 𝟗 𝒙 𝟐 𝒙 ≈ 𝟏𝟑. 𝟏𝟓𝟕𝟔 𝟎. 𝟗 𝒙 𝟐 𝟏 = 𝟐 ′𝐿𝑂𝐺 ′ 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 log 𝟎. 𝟗 𝒙 𝟐 = log 𝑃𝑜𝑤𝑒𝑟 𝑤/ 𝐸𝑥𝑝 𝐿𝑎𝑤 𝟏 𝟐 𝒙 𝟏 log 𝟎. 𝟗 = log 𝟐 𝟐 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝑥 𝟏 log 𝟐 𝒙 = 𝟐 log 𝟎. 𝟗 𝒙 ≈ 𝟏𝟑. 𝟏𝟓𝟕𝟔 SOLVING EXPS USING LOGS METHOD 2 – **NEW** 𝑴𝒆𝒕𝒉𝒐𝒅 𝟐 comes in handy when you have exponentials on both sides with different bases. TWO EXPONENTS, UNLIKE BASES 𝟑𝒙+𝟐 = 𝟒𝟓𝒙 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝐵𝑎𝑠𝑒 ′𝐿𝑂𝐺 ′ 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝒅𝒐𝒏𝒆 𝒍𝒐𝒈 𝟑𝒙+𝟐 = 𝒍𝒐𝒈 𝟒𝟓𝒙 𝑃𝑜𝑤𝑒𝑟 𝑤/ 𝐸𝑥𝑝 𝐿𝑎𝑤 𝐸𝑥𝑝𝑎𝑛𝑑 𝐶𝑜𝑙𝑙𝑒𝑐𝑡 𝑥 ′ 𝑠 𝐹𝑎𝑐𝑡𝑜𝑟 𝑥 ′ 𝑠 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝑥 𝒙 + 𝟐 𝒍𝒐𝒈𝟑 = 𝟓𝒙 𝒍𝒐𝒈 𝟒 𝒙 𝒍𝒐𝒈 𝟑 + 𝟐 𝒍𝒐𝒈 𝟑 = 𝟓𝒙 𝒍𝒐𝒈 𝟒 𝒙 𝒍𝒐𝒈 𝟑 − 𝟓𝒙 𝒍𝒐𝒈 𝟒 = −𝟐 𝒍𝒐𝒈 𝟑 𝒙 𝒍𝒐𝒈 𝟑 − 𝟓 𝒍𝒐𝒈 𝟒 = −𝟐 𝒍𝒐𝒈 𝟑 −𝟐 𝒍𝒐𝒈 𝟑 𝒙 = 𝒍𝒐𝒈 𝟑 − 𝟓 𝒍𝒐𝒈 𝟒 𝒙 ≈ 𝟎. 𝟑𝟕𝟔𝟔𝟗𝟕𝟔𝟔𝟕 TRY IT 𝒙 𝟓𝟐 𝟑𝒙−𝟕 =𝟐 𝒙 ≈ 𝟑. 𝟖𝟎𝟔𝟑𝟒𝟐𝟏𝟏𝟕 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝐵𝑎𝑠𝑒 ′𝐿𝑂𝐺 ′ 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑃𝑟𝑜𝑑𝑢𝑐𝑡 𝐿𝑎𝑤 𝟑 𝟐 𝒙+𝟐 𝒍𝒐𝒈𝟑 𝟐 𝒙+𝟐 =𝟒 𝟓 𝒅𝒐𝒏𝒆 𝟐𝒙 = 𝒍𝒐𝒈𝟒 𝟓 𝟐𝒙 𝒍𝒐𝒈 𝟑 + 𝒍𝒐𝒈 𝟐𝒙+𝟐 = 𝒍𝒐𝒈 𝟒 + 𝒍𝒐𝒈 𝟓𝟐𝒙 𝑃𝑜𝑤𝑒𝑟 𝑤/ 𝐸𝑥𝑝 𝐿𝑎𝑤 𝒍𝒐𝒈 𝟑 + 𝒙 + 𝟐 𝒍𝒐𝒈 𝟐 = 𝒍𝒐𝒈 𝟒 + 𝟐𝒙 𝒍𝒐𝒈 𝟓 𝐸𝑥𝑝𝑎𝑛𝑑 𝒍𝒐𝒈 𝟑 + 𝒙 𝒍𝒐𝒈 𝟐 + 𝟐 𝒍𝒐𝒈 𝟐 = 𝒍𝒐𝒈 𝟒 + 𝟐𝒙 𝒍𝒐𝒈 𝟓 𝐶𝑜𝑙𝑙𝑒𝑐𝑡 𝑥 ′ 𝑠 𝒙 𝒍𝒐𝒈 𝟐 − 𝟐𝒙 𝒍𝒐𝒈 𝟓 = 𝐥𝐨𝐠 𝟒 − 𝐥𝐨𝐠 𝟑 − 𝟐 𝐥𝐨𝐠 𝟐 𝐹𝑎𝑐𝑡𝑜𝑟 𝑥 ′ 𝑠 𝒙 𝒍𝒐𝒈 𝟐 − 𝟐 𝒍𝒐𝒈 𝟓 = 𝒍𝒐𝒈 𝟒 − 𝒍𝒐𝒈 𝟑 − 𝟐 𝒍𝒐𝒈 𝟐 𝐼𝑠𝑜𝑙𝑎𝑡𝑒 𝑥 𝒍𝒐𝒈 𝟒 − 𝒍𝒐𝒈 𝟑 − 𝟐 𝒍𝒐𝒈 𝟐 𝒙 = 𝒍𝒐𝒈 𝟐 − 𝟐 𝒍𝒐𝒈 𝟓 𝒙 ≈ 𝟎. 𝟒𝟑𝟒𝟗𝟔𝟖𝟒𝟓𝟔 TRY IT 𝟐 𝟖 𝟑 𝒙+𝟑 = 𝒙 𝟕(𝟏𝟏)𝟑 𝒙 ≈ −𝟑. 𝟎𝟑𝟔𝟑𝟑𝟗𝟔𝟗𝟗 HOMEWORK Visions 1 P199 5ch P209 3ch