A18 - exp and log fcns - Exponential and Logarithmic Functions Institute of Mathematics University of the Philippines Diliman Lecture 19(IMath UPD

A18 - exp and log fcns - Exponential and Logarithmic...

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Exponential and Logarithmic Functions Institute of Mathematics, University of the Philippines Diliman Lecture 19 (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 1 / 29
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Outline 1 Real Exponents 2 Exponential Functions 3 Equations involving Exponential Expressions 4 Logarithmic Functions (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 2 / 29
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Real Exponents recall: definition and properties of rational exponents (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , then a p < a q . (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , then a p < a q . Ex. 2 3 < 2 4 , (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , then a p < a q . Ex. 2 3 < 2 4 , 3 - 1 > 3 - 2 , (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , then a p < a q . Ex. 2 3 < 2 4 , 3 - 1 > 3 - 2 , 5 1 . 3 < 5 1 . 32 (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , then a p < a q . Ex. 2 3 < 2 4 , 3 - 1 > 3 - 2 , 5 1 . 3 < 5 1 . 32 2 2 ? (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents recall: definition and properties of rational exponents In particular, if a > 1 and p, q with p < q , then a p < a q . Ex. 2 3 < 2 4 , 3 - 1 > 3 - 2 , 5 1 . 3 < 5 1 . 32 2 2 ? We want to define 2 2 such that the property described above holds. (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 3 / 29
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Real Exponents Note that 2 1 . 41421359 ... . If 2 2 is to be defined such that properties of exponents would hold, then: (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 4 / 29
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Real Exponents Note that 2 1 . 41421359 ... . If 2 2 is to be defined such that properties of exponents would hold, then: 2 1 < 2 2 , (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 4 / 29
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Real Exponents Note that 2 1 . 41421359 ... . If 2 2 is to be defined such that properties of exponents would hold, then: 2 1 < 2 2 , 2 1 . 4 < 2 2 , (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 4 / 29
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Real Exponents Note that 2 1 . 41421359 ... . If 2 2 is to be defined such that properties of exponents would hold, then: 2 1 < 2 2 , 2 1 . 4 < 2 2 , 2 1 . 41 < 2 2 , (IMath, UPD) Exponential and Logarithmic Functions Lec. 19 4 / 29
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Real Exponents Note that 2 1 . 41421359 ... . If 2 2 is to be defined such that properties of exponents would hold, then: 2 1 < 2 2 , 2 1 . 4 < 2 2 , 2 1 . 41 < 2 2 , 2 1 . 414 < 2 2 , ...
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