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Lectures and Solutions 3

# Lectures and Solutions 3 - MAT106 Trigonometry...

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MAT106: Trigonometry Justifications/Proofs of Various Properties of Logarithms Below are basic algebraic proofs of the validity of some of the basic properties of logarithms. In order to proceed, the following basic facts should be noted: a. is equivalent to . b. For any logarithmic expression , it is always true that , , and . c. For any positive values and , there exists some value such that . Power Rule Consider the Power Rule: . Note that for some value of . Rewriting the preceding logarithm in exponential form yields . Since , it can be stated that for some value of . So, can be rewritten as ( ) . Thus, by equating exponents (the technique for solving a Type I Exponential Equation), . But, can be rewritten as . Hence, is equivalent to . Therefore, since , the desired result is obtained: .

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Product Rule Consider the Product Rule: . Note that , for some value of . Rewriting the preceding logarithm in exponential form yields . Since , it can be stated that and , for some values of and . So, can be rewritten as . Thus, by equating exponents (the technique for solving a Type I Exponential Equation),
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Lectures and Solutions 3 - MAT106 Trigonometry...

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