MTH405_Wk_3_Lect_2 - MTH405 Wk 3 Lect 2 Mathematical...

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Unformatted text preview: MTH405 Wk 3 Lect 2 Mathematical Functions Exponential Functions Denition. The exponential function f with base a is denoted by y = f (x) = ax where a > 0, a 6= 1, and x is any real number. The numbers a and x are called base and exponent, respectively.The exponential constant e has the value of e = 2.718 . . . and the natural exponential function is dened by f (x) = ex . Theorem 1. (Properties of Exponential Functions) Let a be a positive real number, and let x and y be real numbers with variables, or algebraic expressions. 1. ax · ay = ax+y . x 2. a ay = ax−y . 3. (ax )y = axy . 4. a−x = 1 ax =  1 x . a Exercise. Evaluate f (x) = 2x when x = 3 and when x = −2. Exercise. A particular radioactive element has a half-life of 25 years. For an initial mass of 10 grams, the mass y (in gram) that remains after t years is given by   25t 1 , t ≥ 0. y = 10 2 How much of the initial mass remains after 120 years? 1 Logarithmic Functions Denition. (Denition of Logarithmic function) Let a and x be positive real numbers such that a 6= 1. The logarithm of x with base a is denoted by loga x and is dened as the power to which a must be raised to obtain x. The function f (x) = loga x is the logarithmic function with base a. Exercise. Evaluate each logarithm. 1. log2 8. 2. log2 16. 3. log3 9. 4. log4 2. 5. log4 64. 6. log5 125. 7. log3 3. √ 8. log5 5. 9. log3 1 9  . √ 10. log3 3 3. Exercise. Evaluate each logarithm. 1. log5 1. 2. log10 1 10 . 3. log3 (−1). 4. log4 0. 2 Denition. (Denition of Natural Logarithmic Function) The function f (x) = loge x = ln x, where x > 0 is the Theorem. natural logarithmic function. (Relation between exponential and logarithmic functions) Let a and x be positive real number such that a 6= 1. Then x = am if, and only if, loga x = m. Exercise. Express the following in logarithmic form. 1. 33 = 81. 2. 93 = 719. 3. 36 2 = 6. 1 4. 0.12 = 0.01. Theorem. that (Properties of Logarithms) Let a and x be positive real numbers such a 6= 1. The following properties are true. 1. loga 1 = 0 and ln 1 = 0. 2. loga a = 1 and ln e = 1. 3. loga ax = x and ln ex = x. 4. loga (uv) = loga u + loga v and ln(uv) = ln u + ln v . 5. loga ( uv ) = loga u − loga v and ln( uv ) = ln u − ln v . 6. loga un = n loga u and ln un = n ln u. Exercise. Use the properties above to evaluate each logarithm. 1. log10 100. 2. log10 0.01. 3. ln e2 . 4. ln 1e . Exercise. Express the following as a single logarithm. 1. log 56 + log 14. 2. log 2 + log 3 + log 4. 3. log 68 − log 17. 4. 3 log 4 − log 2. 5. 1 2 log 49 + 6. 1 2 log 100 − log 2. 1 3 log 27. 3 Exercise. Use ln 2 ≈ 0.693, ln 3 ≈ 1.099, and ln 5 ≈ 1.609 to approximate each expression. 1. ln 23 . 2. ln 10. 3. ln 30. Exercise. Use the properties of logarithms to verify that − ln 2 = ln 21 . Exercise. Use the properties of logarithms to expand each expression. 1. log10 7x3 . 2. log6 3. ln √ 8x2 y . 3x−5 . 7 Exercise. Use the properties of logarithms to condense each expression. 1. ln x − ln 3. 2. 1 2 log3 x + log3 5. 3. 3(ln 4 + ln x). 4...
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