Notice in the first graph, khổng lồ the left of the y-axis, e^x increase very slowly, it crosses the axis at y = 1, & to the right of the axis, it grows at a faster và faster rate.The second graph is just the opposite. For negative x"s, the graph decays in smaller and smaller amounts. It crosses the y-axis at y = 1, & then decays at slower & slower rates.The Natural LogThe natural log is the logarithm whose base is e. The two functions, the natural log và the exponential e, are inverses of each other. In other words, saying y = Ln
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For, b"s less than 0, the same occurs except the plots look like the plot of e^-x from above.Exercises1.)Simplify the following expressions.a.)e^(ln 2 + ln x) = 2xb.)ln(e^(1/x)) = 1/x2.)Solve for y.a.)e^(2y) = x^2 = ln(x)b.)ln(y - 1) = x + ln x = xex + 13.)Sketch the following curves on the same axes. Identify the domains of each equation in terms of x.a.)y = ln(-x) and y = -ln(x).
Domain of y = ln(-x): negative infinity to lớn zero (-inf domain of y = -ln(x): zero to lớn positive infinity (0 b.)y = e^(-x) & y = -e^(x).
Domain of y = e-x: negative infinity lớn positive infinity (-inf domain of y = -ex: negative infinity to lớn positive infinity (-inf Application of Exponentials4.)If you invest A dollars at a fixed annual interest rate, r and interest is compounded continuously to your account, the amount of money, Ao, you will have at the kết thúc of t years is,Ao = A e^(rt)Compounded continuously means that the money in your tài khoản is continuously being added interest.
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It can almost be said that the interest is being added every second, day or night.a.)You deposit $621 in an tài khoản that pays 10% interest. How much money will you have after 8 years? after 10 years?8 years: 621e0.8 = $1382.06110 years: 621e = $1688.053b.)How long will it take you to lớn double your money if you invest $500 at an interest rate 6%? = (0.06)/ln(2) = 11.552 yearsView answersReturn to lớn title pageGo khổng lồ next section