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Lecture 4

# MTH 162 Lecture 4: 5.4 General Logarithmic and Exponential Functions Notes Premium

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School
University of Miami
Department
Mathematics
Course
MTH 162
Professor
Pachero
Semester
Fall

Description
MTH 162 Calculus II 5.4 General Logarithmic and Exponential Functions Notes L. Sterling Abstract Provide a generalization to the terms listed in this section. Solving Logarithmic and Exponential Theorems Theorems ▯ log Theorem a – If both x > 0 and a > 0 while a 6= 1, then y = f(x) = log x a and this only happens iﬀ if and only if that y x = a ▯ log is 1 ▯ to ▯ 1 Theorem a – If you let M, N, and even a being all positive real numbers while having a 6= 1, then M = N , and this only happens iﬀ if and only if that loga(M) = log aN) x ▯ a is 1 ▯ to ▯ 1 Theorem – If you let both u and v be real numbers while having both a > 0 and a 6= 1, then u = v and this only happens iﬀ if and only if that u v a = a Solving Logarithmic Equation Example log 4 = 2log x 5 5 2log (x) = log (4) 2 log (x) log (4) 25 = 2 log (x) = log (2) 5 5 x = 2 1 Solving Exponential Equation Example x 3 ▯ 3 = 240 3 ▯ 3 + 3 = 240 + 3 x 3 = 243 3 = 3 5 x = 5 Solving Logarithmic Equation Example ▯1 + log (x + 1) + log (x + 2) = 0 6 6 ▯1 + log 6(x + 1)(x + 2)) = 0 ▯1 + log ((x + 1)(x + 2)) + 1 = 0 + 1 6 log6((x + 1)(x + 2)) = 1 log6((x + 1)(x + 2)) = log6(6) (x + 1)(x + 2) = 6 3 x + 2x + x + 2 = 6 x + 3x + 2 = 6 2 x + 3x + 2 ▯ 6 = 6 ▯ 6 x + 3x ▯ 4 = 0 (x ▯ 1)(x + 4) = 0 x = ▯4; 1 x = 1 Solving Exponential Equation Example x x+3 5 = 4 ▯ ▯ ln(5 ) = ln 4 x+3
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