MTH405_Wk_4_Lect_1

# 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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