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, và to the right of the axis, it grows at a faster & faster rate.The second graph is just the opposite. For negative x"s, the graph decays in smaller và smaller amounts. It crosses the y-axis at y = 1, & then decays at slower và slower rates.The Natural LogThe natural log is the logarithm whose base is e. The two functions, the natural log & 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 lượt thích 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) và y = -ln(x).
Domain of y = ln(-x): negative infinity to lớn zero (-inf domain of y = -ln(x): zero lớn positive infinity (0 b.)y = e^(-x) and y = -e^(x).
Domain of y = e-x: negative infinity to positive infinity (-inf tên miền of y = -ex: negative infinity 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 over of t years is,Ao = A e^(rt)Compounded continuously means that the money in your account 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 trương mục 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 title pageGo khổng lồ next section