**Log**

**= Log - Log**

**Log = b Log****The exponential function can be described as,y = a e^(b x)where a & b are constants. The curve that we use khổng lồ 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 khổng lồ the 0 nguồn 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 và 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, & the second is for a small b. Notice the scales of the plots.**

Bạn đang xem: Exponential review

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.

Xem thêm: Lý Thuyết Chất Dẫn Điện Và Chất Cách Điện - Dòng Điện Trong Kim Loại Và Bài Tập

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

Bạn đang xem: Exponential review

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.

Xem thêm: Lý Thuyết Chất Dẫn Điện Và Chất Cách Điện - Dòng Điện Trong Kim Loại Và Bài Tập

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