20. Exponential Functions A function when the base(a) is some positive number. The exponent is variable(x). The exponential function with base a is defined by:

Example 1 x -2 -1 1 2 f(x) f (x) Domain: Range: x 1 2 f(x) f (x) 4 Domain: 2 Range: x -1 1 2 Horizontal Asymptote:

Example 2 x -2 -1 1 2 f(x) f (x) Domain: Range: x 1 2 f(x) f (x) 9 Domain: 3 Range: x -1 1 2 Horizontal Asymptote:

Example 3 Find the domain, range, and horizontal asymptote. f(x)

Example 4 Find the domain, range, and horizontal asymptote. f(x)

Natural Base, e Special base, e  2.7182818…….. Use a calculator to evaluate the following values of the natural exponential function (round to 5 decimal places):

Compound Interest An investment has its interest compounded n times a year. The amount the investment is worth in t years is given by: where: P = r = n = t =

Example 5 n A(3) What would a $5000 investment be worth in 3 years if the interest rate is 7.5% and the investment is compounded: n A(3) yearly semiannually monthly continuously

21. Logarithmic Functions Exponential functions f (x) = ax are one-to-one functions. This means they each have an inverse function. We denote the inverse function with loga, the logarithmic function with base a.

Definition logax is Switch from logarithmic form to exponential form:

Evaluating logarithms Switch from exponential form to logarithmic form:

Graph y Create a table of points: 6 x 1/2 1 2 4 1 x -1 1 -6

Graph New domain restriction: No negative under an even root y Domain: Range: Vertical Asymptote: x 1 New domain restriction: No negative under an even root No division by zero Only

Properties 1. loga1 = 0 (you must raise a to the power of 0 in order to get a 1) 2. logaa = 1 (you must raise a to the power of 1 to get an a) 3. logaax = x (you must raise a to the power of x to get ax) 4. alogax = x (logax is the power to which a must be raised to get x)

Common Logarithm (Base 10) With calculator: Without calculator:

Natural Logarithm (Base e) With calculator: Without calculator:

22. Laws of Logarithms 1. loga(AB) = loga A + loga B Let A, B, and C be any real numbers with A > 0 and B > 0. 1. loga(AB) = loga A + loga B 2. loga(A/B) = loga A - loga B 3. loga(AC) = C loga A

Applying the laws Use the laws of logarithms to expand the following:

Applying the laws - continued Use the laws of logarithms to combine the following:

Change of Base To evaluate other bases on the calculator, use the following formula: loga b

23. Solving equations Isolate exponential function and apply logarithm function to both sides of the equation. Isolate the logarithm function and apply the base to both sides of the equation. Remember logarithm laws and inverse properties:

Example 1

Example 2

Example 3

Example 4

Example 5

24. Exponential Applications Growth: n(t) = n0ert, positive power (for population models) Decay: m(t) = m0e-rt, negative power (for decay models) Cooling: T(t)=Ts+D0e -rt, negative power (indicates loss in difference between object and surrounding temperature) Logarithms: pH scale, earthquake intensity, decibel levels

Example 1 (a) What is the initial number of bacteria? (b) What is the relative rate of growth? Express your answer as a percentage. (c) How many bacteria are in the culture after 5 hours? Please round the answer to the nearest integer. (d) When will the number of bacteria reach 10,000? Please round the answer to the nearest hundredth.

Example 2 Find a function that models the population t years after 2005? (b) Find the projected population in the year 2016? Please round the answer to the nearest thousand. (c) In what year will the population reach 200000?

Example 3 (a) How much remains after 60 days? (b) When will 10 grams remain? Please round the answer to the nearest day.

Half-life for radioactive isotopes (c) Find the half-life of polonium-210. In general: m(t) = m0e-rt

Example 4 The burial cloth of an Egyptian mummy is estimated to have 56% of the carbon-14 it contained originally. How long ago was the mummy buried? Round answer to nearest ten. (Carbon-14 has a half-life of 5730 years.)