5.1 Exponential Functions Rules For Exponents If a > 0 and b > 0, the following hold true for all real numbers x and y.

5.1 Exponential Functions If we apply the quotient rule, we get:

5.1 Exponential Functions For any nonzero number x: and

5.1 Exponential Functions Examples:

5.1 Exponential Functions Examples: (5x-2)3 = 125x-6=125/x6 (3x/y3)2 = 9x2/y6 (4x)-1 = 1/(4x) (2a3b-3c4)3 = 8a9b-9c12 40 = 1 2-1 = ½ (½)-2 = 4 5-2 = 1/25

5.1 Exponential Functions Simplify:

5.1 Exponential Functions

5.1 Exponential Functions Simplify: Rewrite: Notice:

5.1 Exponential Functions Simplify: Rewrite: Notice:

5.1 Exponential Functions Simplify: Rewrite: Notice:

5.1 Exponential Functions Then If Examples Since Since Since

5.1 Exponential Functions In general, if n is a multiple of m, then If n is odd If n is even

5.1 Exponential Functions Use the rules for exponents to solve for x 4x = 128 (2)2x = 27 2x = 7 x = 7/2 2x = 1/32 2x = 2-5 x = -5

5.1 Exponential Functions 27x = 9-x+1 (33)x = (32)-x+1 33x = 3-2x+2 3x = -2x+ 2 5x = 2 x = 2/5 (x3y2/3)1/2 x3/2y1/3

5.1 Exponential Functions Definition Exponential Function Let a be a positive real number other than 1, the function f(x) = ax is the exponential function with base a.

5.1 Exponential Functions 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x If b > 1, then the graph of b x will: Rise from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) y = 2 x

5.1 Exponential Functions 4 3 2 1 -2 -3 -4 -5 y -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 x If 0 < b < 1, then the graph of b x will: Fall from left to right. Not intersect the x-axis. Approach the x-axis. Have a y-intercept of (0, 1) y = (1/2) x

5.1 Exponential Functions Natural Exponential Function where e is the natural base and e  2.718…

5.1 Exponential Functions f(x) = 2x h(x) = (0.5)x g(x) = ex Domain Range Increasing or Decreasing Point Shared On All Graphs (-∞, ∞) (-∞, ∞) (-∞, ∞) (0, ∞) (0, ∞) (0, ∞) Inc. Dec. Inc. (0, 1)

5.1 Exponential Functions Use translation of functions to graph the following. Determine the domain and range f (x) = 2(x + 2) – 3 Domain (-∞, ∞) Range (-3, ∞)

5.1 Exponential Functions Definitions Exponential Growth and Decay The function y = k ax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. y new amount yO original amount b base t time h half life

5.1 Exponential Functions An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g. Find the amount remaining after t hours. Find the amount remaining after 60 hours. a. y = yobt/h y = 2 (1/2)(t/15) b. y = yobt/h y = 2 (1/2)(60/15) y = 2(1/2)4 y = .125 g

5.1 Exponential Functions A bacteria double every three days. There are 50 bacteria initially present Find the amount after 2 weeks. When will there be 3000 bacteria? a. y = yobt/h y = 50 (2)(14/3) y = 1269 bacteria

5.1 Exponential Functions A bacteria double every three days. There are 50 bacteria initially present When will there be 3000 bacteria? b. y = yobt/h 3000 = 50 (2)(t/3) 60 = 2t/3

5.2 Simple and Compound Interest Formulas for Simple Interest Suppose P dollars are invested at a simple interest rate r, where r is a decimal, then P is called the principal and P ·r is the interest received at the end of one interest period.

5.2 Simple and Compound Interest Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r is given by the two formulas below. 1. For n compoundings per year: 2. For continuous compounding:

5.2 Simple and Compound Interest Find the balance after 10 years if $1000.00 is invested at 4% and the account pays simple interest.

5.2 Simple and Compound Interest Find the balance after 10 years if $1000.00 is invested at 4% and the interest is compounded: a. Semiannually $1485.95 b. Monthly: $1490.83 c. Continuously: $1491.82

5.3 Effective Rate and Annuities Effective Annual Rate The effective annual rate of ieff of APR compounded k times per year is given by the equation Another name for effective annual rate is effective yield

5.3 Effective Rate and Annuities What is the better rate of return, 7% compounded quarterly or 7.2 % compounded semianually?

5.3 Effective Rate and Annuities 1.071 – 1 = .071 = 7.1% 1.073 – 1 = .073 = 7.3% 7.2% compounded semiannually is better.

5.3 Effective Rate and Annuities What is the better rate of return, 8 % compounded monthly or 8.2 % compounded quarterly?

5.3 Effective Rate and Annuities 8.3% 8.5% 8.2% quarterly is better.

5.3 Effective Rate and Annuities Future Value of an Ordinary Annuity The Future Value S of an ordinary annuity consisting of n equal payments of R dollars, each with an interest rate i per period is

5.3 Effective Rate and Annuities Suppose $25.00 per month is invested at 8% compounded quarterly. How much will be in the account after one year? 1st quarter $25.00 2nd quarter $25.00(1+.08/4)+ $25.00 = $50.50 3rd quarter $50.50(1+.08/4)+ $25.00 = $76.51 4th quarter $76.51(1+.08/4) + $25.00 = $103.04

5.3 Effective Rate and Annuities Present Value of an Ordinary Annuity The Present Value A of an ordinary annuity consisting of n equal payments of R dollars, each with an interest rate i per period is

5.4 Logarithmic Functions The inverse of an exponential function is called a logarithmic function. Definition: x = a y if and only if y = log a x

5.4 Logarithmic Functions

5.4 Logarithmic Functions Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f. Domain: (0, ∞) Range: (-∞, ∞)

5.4 Logarithmic Functions The function f (x) = log a x is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote: x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)

5.4 Logarithmic Functions Find the inverse of g(x) = 3x. (1,3) (0,1) (-1,1/3) (3,1) Note: The function and it’s inverse are symmetrical about the line y = x. (1,0) (1/3,-1)

5.4 Logarithmic Functions Find the inverse of g(x) = ex. ln x is called the natural logarithmic function

5.4 Logarithmic Functions So So So So

5.4 Logarithmic Functions loga(ax) = x for all x   alog ax = x for all x > 0 loga(xy) = logax + logay loga(x/y) = logax – logay logaxn = n logax Common Logarithm: log 10 x = log x Natural Logarithm: log e x = ln x All the above properties hold.

5.4 Logarithmic Functions Product Rule

5.4 Logarithmic Functions Quotient Rule

5.4 Logarithmic Functions Power Rule

5.4 Logarithmic Functions Expand

5.4 Logarithmic Functions Find an equation of best fit for the data (1,3), (2,12), (3,27), (4,48)

5.5 Graphs of Logarithmic Functions The function f (x) = log a x is called a logarithmic function. Domain: (0, ∞) Range: (-∞, ∞) Asymptote: x = 0 Increasing for a > 1 Decreasing for 0 < a < 1 Common Point: (1, 0)

5.5 Graphs of Logarithmic Functions The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula. where b is any other appropriate base. (usually base 10 or base e)

5.5 Graphs of Logarithmic Functions 3 5 1 11 2 29 Sketch the graph of Domain (2,) Range (-, )

5.5 Graphs of Logarithmic Functions Sketch the graph of Domain (-2,) Range (-, )

5.5 Graphs of Logarithmic Functions Sketch the graph of Domain (-3,) Range (-, )

5.5 Graphs of Logarithmic Functions On the Richter scale, the magnitude R of an earthquake can be measured by the intensity model. R = Magnitude a = Amplitude T = Period B = Damping Factor

5.5 Graphs of Logarithmic Functions What is the magnitude on the Richter scale of an earthquake if a = 300, T = 30 and B = 1.2?

5.6 Solving Exponential Equations Solve: 4 3x = 16 x – 2 The bases can be rewritten as: (22) 3x = (24) (x – 2) 2 6x = 2 4x – 8 6x = 4x – 8 2x = -8 x = -4

5.6 Solving Exponential Equations To solve exponential equations, pick a convenient base (often base 10 or base e) and take the log of both sides. Solve:

5.6 Solving Exponential Equations Take the log of both sides: Power rule:

5.6 Solving Exponential Equations Solve for x: Divide:

5.6 Solving Exponential Equations To solve logarithmic equations, write both sides of the equation as a single log with the same base, then equate the arguments of the log expressions. Solve:

5.6 Solving Exponential Equations Write the left side as a single logarithm:

5.6 Solving Exponential Equations Equate the arguments:

5.6 Solving Exponential Equations Solve for x:

5.6 Solving Exponential Equations

5.6 Solving Exponential Equations Check for extraneous solutions. x = -3, since the argument of a log cannot be negative

5.6 Solving Exponential Equations To solve logarithmic equations with one side of the equation equal to a constant, change the equation to an exponential equation Solve:

5.6 Solving Exponential Equations Write the left side as a single logarithm:

5.6 Solving Exponential Equations Write as an exponential equations:

5.6 Solving Exponential Equations Solve for x: