L> ReviewExponential nhận xét Base eThe number e has been called one of the most important numbers in all of mathematics. However, it is important lớn remember that e is just a number. Calculated to nine decimal places,e = 2.718281828e can be extended to lớn countless decimal places and no patterns have ever been discovered. In this sense e, is very similar khổng lồ pi.Look at these two graphs. The first is the graph of y = e^x and the second is y= e^-x  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 is the same as e^y = x. Look at the plot of y = Ln.The logarithm grows fast at first, then gradually slows. It also crosses the x-axis at 1 và can only be found for x > 0. Therefore, Log<1> = 0, Ln<0 PropertiesHere are some important properties of exponential & log functions, you may find useful.e^(a + b) = e^a * e^be^(a - b) = e^a / e^b(e^a)^b = e^(a * b)Log = Log + LogLog = Log - LogLog = b LogThe exponential function can be described as,y = a e^(b x)where a và b are constants. The curve that we use to lớn fit data sets is in this size so it is important lớn understand what happens when a and b are changed.Recall that any number or variable when raised to lớn the 0 power is 1. In this case if b or x is 0 then, e^0 = 1. So at the y-intercept or x = 0, the function becomes y = a * 1 or y = a. Therefore, the constant a is the y-intercept of the curve.The other parameter in our equation is b. If b is very small & greater than 0, the function flattens out. The curve increases at a slower rate then for large b"s. On the contrary, for large b"s the curve increases quickly. Look at these two plots. The first is for an equation with a large b, and the second is for a small b. Notice the scales of the plots.

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Natural log  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