Exponential and Logarithmic Functions 1 2 1 Exponential

Exponential and Logarithmic Functions 1

2 -1 Exponential Functions Function type linear quadratic polynomial exponential Example f(x) = 3 x + 2 f(x) = 3 x 2 + 2 x - 3 f(x) = 2 x 4 + 3 x 2 f(x) = 2 x f(x) = 2. 63 x-1 2

Graph Exponential Functions (b > 1) • Graph y = 2 x for x = -3 to 3 x y -3 1/8 -2 1/4 -1 1/2 0 1 1 2 2 4 3 8

Graph: f(x) = 2 x 4

Graph: f(x) = 2 -x

Graph Exponential Function (0< b < 1) • Graph y = (1/2)x for x = -3 to 3 x y -3 8 -2 4 -1 2 0 1 1 1/2 2 1/4 3 1/8

• 2 is the base of the exponential –“exponential base 2” • the base can be any positive number • common bases are 2, 10, and e Exponential functions with base e f(x) = ex • e is a real number constant (like ) value = 2. 7182818… • frequently seen as the base for exponential functions • called the natural base 7

Properties of the exponential functions: • f(x) = bx and f(x) = b-x, - < x < • b is called the base • can always make b > 1 • e. g. f(x) = = 2 -x • domain: (- , ) range: (0, ) • continuous • f(x) = bx is an increasing function • f(x) = b-x is a decreasing function

• y-intercept: y = 1 • no x-intercepts; graph always above x axis • x-axis is an asymptote: as x + for f(x) = b-x as x - for f(x) = bx • bx = by x = y (one-to-one property)

Growth Decay

Exponential function – A function of the form y=abx, where b>0 and b 1. Step 1 – Make a table of values for the function.

Now that you have a data table of ordered pairs for the function, you can plot the points on a graph. (-2, 1/9) (0, 1) (2, 9) Draw in the curve that fits the plotted points. y y x x

APPLICATION Simple Interest formulas I = Prt A = P + Prt = P(1 + rt)

P = principal invested (also called present value) r = annual interest rate (expressed as a decimal) t = time in years I = interest earned A = total amount after t years (also called future value)

Simple Interest You invest $100. 00 for at 10% simple interest. How much do you have at the end of 2 years? We can do this in our heads: 10% of $100 is $10 that's $10 interest earned per year for 2 years for a total of $20 interest plus the $100 original investment. . . for a new amount of : $120

P = ($100) r = (10%) t = (2 years) I = ($20) A = ($120. 00)

If you know any 3 of the variables, the formula (plus some algebra) can be used to solve for the 4 th variable: Example: $100 is invested (simple interest) for 10 years, and the investment doubled in value. What was the interest rate? The equation: 200 = 100(1 + r(10)) Solve: r = 0. 10= (10%)

You deposit $1500 in an account that pays 2. 3% interest compounded yearly, 1) What was the initial principal (P) invested? 2) What is the growth rate (r)? The growth factor? 3) Using the equation A = P(1+r)t, how much money would you have after 2 years if you didn’t deposit any more money? 1) The initial principal (P) is $1500. 2) The growth rate (r) is 0. 023. The growth factor is 1. 023.

2 -2 Logarithmic Functions The common log of a number is that exponent (or power) to which 10 must be raised to obtain the number. Notation: y = log (x) or y = log x "y = log of x" Example: • log (1000) =. . . • . . . the power to which 10 must be raised to obtain 1000 • so log(1000) = 3 19

Another way to put it: 3 = log(1000) because 103 = 1000 y = log(x) means 10 y = x Be CAREFUL! log x + 2 log(x + 2) = log(x) + 2 20

Graphing a log function y = log 2 x: x 1 4 8 1/2 y 0 2 3 -1 21

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+ Properties of logs Property 1 Recall: if m = 102 then 2 = log m 100000 = 100 x 1000 mn = m x n 105 = 102 x 103 10 log mn = 10 log m x 10 log n 10=mlog mn 10 log exponents) +log mn m = log then x = y) the log of a product = the sum of the logs 23

- Property 2 ================ 100000 1000 m/n = m n 102 = 105 103 10 log m/n = 10 log m 10 log n m m/n 10 log 10=log exponents) - log m/n m = log then x = y) the log of a quotient = the difference of the logs =====================24

Property 3 log mr = log mm … m (for r factors) = log m +. . . + log m (for r terms) = r log mr = r log m the log of a power = the exponent times the log of the base 25

These relationships hold for any base: 1. loga mn = loga m + loga n (log of a product) 2. loga m/n = loga m - loga n (log of a quotient) 3. loga mr = r loga m (log of a power) Each property can be used in two directions, e. g. log (10)(20) = log 10 + log 20 uses property 1 going from left-hand to right-hand side called expansion ( ) 26

log 10 - log 20 = log (10/20) uses property 2 going from right-hand to lefthand side called collection, or writing as the log of a single expression “log expression” ( ) the book directions are “write as a one logarithm” - this is ambiguous. Read “Write as the log of a single expression” 3 log x not acceptable as an answer, but log x 3 27

Examples: Write the equivalent exponential equation and solve for y. Logarithmic Equation y = log 216 y = log 2( ) Equivalent Exponential Equation Solution 16 = 2 y 16 = 24 y = 4 = 2 y = 2 -1 y = – 1 y = log 416 16 = 4 y 16 = 42 y = 2 y = log 51 1=5 y 1 = 50 y = 0 28

Properties of Logarithms 1. loga 1 = 0 since a 0 = 1. 2. loga a = 1 since a 1 = a. 3. loga ax = x and alogax = x inverse property 4. If loga x = loga y, then x = y. one-to-one property Examples: Solve for x: log 6 6 = x log 6 6 = 1 property 2 x = 1 Simplify: log 3 35 = 5 property 3 Simplify: 7 log 79 = 9 property 3 29

Properties of Natural Logarithms 1. ln 1 = 0 since e 0 = 1. 2. ln e = 1 since e 1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property Examples: Simplify each expression. inverse property 2 property 1 30