MTH405_Wk_4_Lect_1 - MTH405 Wk 4 Lect 1 Change of Base...

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Unformatted text preview: MTH405 Wk 4 Lect 1 Change of Base Formula Let a, b and x be positive real numbers such that a 6= 1 and b 6= 1. Then loga x = ln x . ln a Exercise. 1. Use natural logarithms to evaluate log3 5. 2. Use natural logarithms to evaluate log6 2. Solving Exponential and Logarithmic equations (Properties of Exponential and Logarithmic equations) Let a be a positive real number such that a 6= 1, and let x and y be real numbers. Then the following properties are true. Theorem. 1. ax = ay if, and only if, x = y . (a) loga x = loga y if, and only if, x = y (x > 0, y > 0). (b) loga (ax ) = x and ln(ex ) = x. (c) a(loga x) = x and e(ln x) = x. Exercise. Use the properties above to solve each equation. 1. 4x+2 = 64. 2. ln(2x − 3) = ln 11. 3. 2 log x = log 16. 4. 1 2 log x = log 6. 5. logx 25 = 2. 6. x log2 8 = 6. 7. log4 x = 2. 8. log5 x = 3. 1 Solving exponential equations To solve an exponential equation, rst isolate the exponential expression. Then take the logarithm of each side of the equation and solve for the variable. Exercise. Solve each exponential expression. 1. 2x = 7. 2. 4x−3 = 9. 3. 2ex = 10. 4. 5 + ex+1 = 20. Solving Logarithmic Equations To solve a logarithmic equation, rst isolate the logarithmic expression. Then exponentiate each side of the equation and solve for the variable. Exercise. Solve each logarithmic equation. 1. 2 log4 x = 5. 2. 1 4 log2 x = 12 . 3. 3 log10 x = 6. 4. log3 2x − log3 (x − 3) = 1. Exercise. A deposit of $5000 is placed in a savings account for 2 years. The interest on the account is compounded continuously. At the end of 2 years, the balance in the account is $5416.44. What is the annual interest rate for this account? r n ) where P =initial amount, A =nishing ( Hint: Amount A = P (1 + 100 amount, r =annual interest rate, n =no. of periods of the interest ) 2 ...
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