1. Section 6.3

2. This value is so important in mathematics that it has been given its own symbol, e, sometimes called Euler’s number. The number e has many of the same characteristics as π. Its decimal expansion never terminates or repeats in a pattern. It is an irrational number. To eleven decimal places, e = 2.71828182846 Value of e

3.  The base e, which is approximately e = 2.718281828… is an irrational number called the natural base.

4. Use your calculator to evaluate the following. Round our answers to 4 decimal places. 7.3891 0.1353 1.3499 103.0455

5. The function f, represented by f(x) = ce x is the natural exponential function, where c is the constant, and x is the exponent.

6. Properties of an natural exponential function: Domain: (-∞, ∞) Domain: (-∞, ∞) Range: (0, ∞) Range: (0, ∞) y-intercept is (0,c) y-intercept is (0,c) f increases on (-∞, ∞) f increases on (-∞, ∞) The negative x-axis is a horizontal asymptote. The negative x-axis is a horizontal asymptote. f is 1-1 (one-to-one) and therefore has an inverse. f is 1-1 (one-to-one) and therefore has an inverse. Example f(x) = ce x

7. State the transformation of each function, horizontal asymptote, y-intercept, and domain and range for each function. 1 unit right, down 3 units h.a. y = -3 y-int: f(0) = -2.6 Domain: (-∞, ∞) Range: (-3, ∞). reflect x-axis h.a. y = 0 y-int: f(0) = -1 Domain: (-∞, ∞) Range: (-∞, 0). reflect y-axis, down 5 h.a. y = -5 y-int: f(0) = -4 Domain: (-∞, ∞) Range: (-5, ∞).

8. The function of the form P(t) = P 0 e kt Models exponential growth if k > 0 and exponential decay when k < 0. T = time P 0 = the initial amount, or value of P at time 0, P > 0 k = is the continuous growth or decay rate (expressed as a decimal) e k = growth or decay factor

9. For each natural exponential function, identify the initial value, the continuous growth or decay rate, and the growth or decay factor.. Initial Value : 100 Growth Rate: 2.5% Growth Factor: = 1.0253 Initial Value : 500 Decay Rate: -7.5% Decay Factor: = 0.9277

10.  Ricky bought a Jeep Wrangler in 2003. The value of his Jeep can by modeled by V(t)=25499e -0.155t where t is the number of years after 2003. a) Find and interpret V(0) and V(2). a) What is the Jeep’s value in 2007?

11. What is the Natural Logarithmic Function? Logarithmic Functions with Base 10 are called “common logs.”Logarithmic Functions with Base 10 are called “common logs.” log (x) means log 10 (x) - The Common Logarithmic Functionlog (x) means log 10 (x) - The Common Logarithmic Function Logarithmic Functions with Base e are called “natural logs.”Logarithmic Functions with Base e are called “natural logs.” ln (x) means log e (x) - The Natural Logarithmic Functionln (x) means log e (x) - The Natural Logarithmic Function

12.  Let x > 0. The logarithmic function with base e is defined as y = log e x. This function is called the natural logarithm and is denoted by y = ln x.  y = ln x if and only if x=e y.

13. ln (1) ln (e) ln (e x ) ln (1) = log e (1) = 0 since e 0 = 1 ln(e) = log e (e) = 1 since 1 is the exponent that goes on e to produce e 1. ln (e x ) = log e e x = x since e x = e x = x

14. Evaluate the following.

15. The graph of y = lnx is a reflection of the graph of y =e x across the line y = x.

16. Domain: (0, ∞)Domain: (0, ∞) Range: (-∞, ∞)Range: (-∞, ∞) x-intercept is (1,0)x-intercept is (1,0) Vertical asymptote x = 0.Vertical asymptote x = 0. f is 1-1 (one-to-one)f is 1-1 (one-to-one) f(x) = ln x

17. For each function, state the transformations applied to y = lnx. Determine the vertical asymptote, and the domain and range for each function. b. f(x) = ln(x-4) + 2c. y = -lnx - 2 4 Right, Shift Up 2 V.A. x = 4 Domain: (4, ∞) Range: (-∞, ∞) Reflect x axis down 2 V.A. x = 0 Domain: (0, ∞) Range: (-∞, ∞)

18. Find the domain of each function algebraically. (31, ∞ ) (-∞, 2.7 ) f(x) = ln (x-31) f(x) = ln (5.4 - 2x) + 3.2