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Section 16-1: Exponential Expressions, Equations, and Formulas Learning Outcome 1 nt r Use the formula A = P 1 + to find the compound amount (total paid back) for a loan of n $10,000, for a term of 5 years at an annual interest rate of 8.4% with the interest compounded semiannually. r A = P 1 + n nt 0.084 A = 10,000 1 + 2 Substitute in the formula. P = $10,000; r = 0.084; n = 2; t =5 2( 5 ) A = 10,000(1 + 0.042)10 A = 10,000(1.042)10 A = 10,000(1.508958131) A = $15,089.58 Simplify expression inside parentheses and exponent. Further simplify expression inside parentheses. Simplify power. Perform multiplication. The amount to be paid back is $15,089.58. Learning Outcome 2 The formula A = Pe rt is the formula used to find the compound amount when the compounding is continuous. A is the compound amount; P is the principal, r is the annual interest rate, and t is the number of years of the loan or investment. If $10,000 is loaned with continuous compounding for 5 years at 8.4% annual interest, what is the compound amount? A = Pen Substitute appropriate values in formula. 0.084(5) A = 10,000e Simplify the exponent. A = 10,000e0.42 Use a calculator to evaluate the formula. A = $15,219.62 Learning Outcome 3 Solve the equation: 93x+2 = 272 93x+2 = 272 Write both sides of the equation with the same base. 2 3x + 2 3 2 (3 ) = (3 ) Multiply the exponents to simplify. Perform the multiplication. 32(3x+2) = 33(2) 36x+4 = 36 If the bases are the same, the exponents will be equal. Write the two exponents as an equation. Solve the equation for x. 6x + 4 = 6 6x = 6 − 4 6x = 2 2 x= 6 1 x= 3 For x = 1 the original equation will be true. 3 Section 16-2: Logarithmic Expressions Learning Outcome 1 Write 81 = 34 in logarithmic form. It is necessary to recognize the pattern that 3 is the base, 4 is the exponent, and 81 is the power. Thus, in logarithmic form we have: log 3 81 = 4. Learning Outcome 2 Write the expression log 15 225 = 2 in exponential form. We note that 15 is the base, 2 is the exponent, and 225 is the power. Thus, we have 152 = 225. Learning Outcome 3 Use a calculator to find log 42. log 42 = 1.62324929 Learning Outcome 4. Use your calculator to evaluate log 7 343. log a log 343 . log 7 343 = =3 Use calculator and log function by applying the rule logb a = log 7 log b Verify the result by raising 7 to the 3rd power. 7(7)(7) = 343. Learning Outcome 5 I to find the decibel rating for a sound that is 200 times the I0 Use the formula bel = 10 log threshold sound (200I0). I I0 200I0 bel = 10 log I0 bel = 10 log bel = 10 log (200) bel = 23.01 Write the formula, and substitute for I. I0 cancels. Find log 200 and multiply by 10. Learning Outcome 6 Use the laws of exponents to evaluate the following logarithm expressions. log (5)(12) log (5)(12) = log 5 + log 12 Rewrite the indicated multiplication as addition. log (5)(12) = 0.6989700043 + 1.079181246 = 1.77815125 Rewrite as the sum of two logarithms: ln (3)(5) ln (3)(5) = ln (3) + ln (5) ln (3)(5) = 1.098612289 + 1.609437912 = 2.708050201