Unit 3 Exponential and Logarithmic Functions Exponential Functions

Unit 3 Exponential and Logarithmic Functions

Exponential Functions – Alg 2: 4. 1 • Vocabulary – Exponential Function y=abx – Base (Common Ratio)- factor of change – Growth (Appreciation) – Decay (Depreciation) – Asymptote – line that a function approaches but never touches – Inverse Function (4. 2) – Logarithmic Function (4. 3 -4. 5) – Natural Log/Base e (4. 6)

Determine Growth vs Decay • Given y=abx • a= y-intercept (starting amount, initial amount) • b= growth or decay factor – If b > 1 = growth – If 0 < b < 1= decay

Graphing Exponential Functions • • Make a Table of Values Enter values of X and solve for Y Plot on Graph Examples: – Y = 2 x state D: – Y = (12)x state D: and R:

You can model growth or decay by a constant percent increase or decrease with the following formula: In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor.

Appreciation - Growth • Amount of function is INCREASING – Growth! • A(t) = a * (1 + r)t – A(t) is final amount – a is starting amount – r is rate of increase – t is number of years (x) • Example: Invest $10, 000 at 8% rate – when do you have $15, 000 and how much in 5 years?

Depreciation - Decay • Amount of function is DECREASING – Decay! • A(t) = a * (1 - r)t – A(t) is final amount – a is starting amount – r is rate of decrease – t is number of years (x) • Example: Buy a $20, 000 car that depreciates at 12% rate – when is it worth $13, 000 and how much is it worth in 8 years?

Compounding Interest • Interest is compounded periodically – not just once a year • Formula is similar to appreciation/depreciation – Difference is in identifying the number of periods • A(t) = a ( 1 + r/n)nt – A(t), a and r are same as previous – n is the number of periods in the year

Examples of Compounding • You invest $750 at the 11% interest with different compounding periods for 1 yr, 10 yrs and 30 yrs: – 11% compounded annually – 11% compounded quarterly – 11% compounded monthly – 11% compounded daily

Continuous Compounding • Continuous compounding is done using e – e is called the natural base – Discovering e – compounding interest lab • Equation for continuous compounding – A(t) = a*ert • A(t), a, r and t represent the same values as previous • Example: $750 at 11% compounded continuously

Inverse Functions • Reflection of function across line x = y • Equivalent to switching x & y values • Example:

Example Graph the relation and connect the points. Then graph the inverse. Identify the domain and range of each relation. x 1 3 4 5 6 y 0 1 2 3 5 Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair. x y 0 1 1 3 2 4 3 5 5 6 • • •

Example Continued Reflect each point across y = x, and connect them. Make sure the points match those in the table. • • Domain: {1 ≤ x ≤ 6} Range : {0 ≤ y ≤ 5} Domain: {0 ≤ y ≤ 5} Range : {1 ≤ x ≤ 6} • • •

Inverse Functions • Steps for creating an inverse 1. 2. 3. 4. Rewrite the equation from f(x) = to y = Switch variables (letters) x and y Solve equation for y (isolate y again) Rewrite new function as f-1(x) for new y

Inverse Functions • Once inverse is set up, use opposite operations – If adding – subtract – If subtracting – add – If multiplying – divide – If dividing – multiply

Undo operations in the opposite order of the order of operations. Helpful Hint The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses

Inverse Functions - Examples • Example: f(x) = 2 x – 3 • Example: f(x)=

Example: Retailing Applications Juan buys a CD online for 20% off the list price. He has to pay $2. 50 for shipping. The total charge is $13. 70. What is the list price of the CD? Step 1 Write an equation for the total charge as a function of the list price. c = 0. 80 L + 2. 50 Charge c is a function of list price L.

Example Continued Step 2 Find the inverse function that models list price as a function of the change. c – 2. 50 = 0. 80 L Subtract 2. 50 from both sides. c – 2. 50 = L 0. 80 Divide to isolate L. Beware: Don’t change variables (letters) in word problems – they mean or represent something

Example Continued Step 3 Evaluate the inverse function for c = $13. 70. Substitute 13. 70 – 2. 50 L= for c. 0. 80 = 14 The list price of the CD is $14. Check c = 0. 80 L + 2. 50 = 0. 80(14) + 2. 50 = 11. 20 + 2. 50 = 13. 70 Substitute.

Logarithms – Alg. 2: Chap 4. 3 • Inverse of an exponential function • Logbx = y – b is the base (same as exponential function) – Converts to: by = x • From exponential function: bx = y • Write logarithmic function: logby = x • If there is no base indicated – it is base 10 • Example: log x = y

You can write an exponential equation as a logarithmic equation and vice versa. Reading Math Read logb a= x, as “the log base b of a is x. ” Notice that the log is the exponent.

Example 1: Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 35 = 243 log 3243 = 5 1 2 The base of the exponent becomes the base of the logarithm. 1 2 25 = 5 log 255 = 104 = 10, 000 log 1010, 000 = 4 6– 1 = ab = c 1 6 log 6 1 = – 1 6 logac =b The exponent is the logarithm. An exponent (or log) can be negative. The log (and the exponent) can be a variable.

Example 2: Converting from Logarithmic to Exponential Form Write each logarithmic form in exponential equation. Logarithmic Form Exponential Equation 1 log 99 = 1 9 =9 log 2512 = 9 29 = 512 log 82 = log 4 1 16 1 3 = – 2 logb 1 = 0 The base of the logarithm becomes the base of the power. The logarithm is the exponent. 1 3 8 =2 4– 2 = b 0 = 1 1 16 A logarithm can be a negative number. Any nonzero base to the zero power is 1.

Solving & Graphing Logarithms • • Write out in exponential form: b? = x What value needs to go in for ? Example: log 327 = ? Graphing – – Plot out the Exponential Function – Table of values – Switch the x and y coordinates – Domain of exponential is range of logarithm (limits) – Range of exponential is domain of logarithm (limits) • Example: Plot 2 x and then log 2 x

Properties of Logarithms Alg 2: 4. 4 • Product Property: logbx + logby = logb(x*y) – Example: log 64 + log 69 • Quotient Property: logbx – logby = logb(x/y) – Example: log 5100 – log 54 • Power Property: logbxy = y*logbx – Examples: log 104 log 5252

Example 1: Product Property Express log 64 + log 69 as a single logarithm. Simplify. log 64 + log 69 log 6 (4 9) To add the logarithms, multiply the numbers. log 6 36 Simplify. 2 Think: 6? = 36.

Example 2: Quotient Property Express log 5100 – log 54 as a single logarithm. Simplify, if possible. log 5100 – log 5 4 log 5(100 ÷ 4) To subtract the logarithms, divide the numbers. log 525 Simplify. 2 Think: 5? = 25.

Example 3 – Power Property Express as a product. Simplify, if possibly. a. log 104 b. log 5252 4 log 10 4(1) = 4 2 log 525 Because 101 = 10, log 10 = 1. 2(2) = 4 Because 52 = 25, log 525 = 2.

More Logarithmic Properties • Inverse Property: logbbx = x & blogbx = x – Example: log 775 – Example: 10 log 2 • • Change of Base: logbx = Example: log 48 Example: log 550 Change of base in Calculator: MATH, log. BASE( – log--- : type base in first then value

Example: Geology Application The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9. 3 How many times as much energy did this earthquake release compared to the 6. 9 -magnitude earthquake that struck San Francisco in 1989? The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula. Substitute 9. 3 for M.

Example Continued Multiply both sides by 23. æ E ö 13. 95 = log ç 11. 8 ÷ è 10 ø Simplify. Apply the Quotient Property of Logarithms. Apply the Inverse Properties of Logarithms and Exponents.

Example Continued Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the tsunami was 5. 6 1025 ergs.

Example Continued Substitute 6. 9 for M. Multiply both sides by 23. Simplify. Apply the Quotient Property of Logarithms.

Example Continued Apply the Inverse Properties of Logarithms and Exponents. Given the definition of a logarithm, the logarithm is the exponent. Use a calculator to evaluate. The magnitude of the San Francisco earthquake was 1. 4 1022 ergs. 25 5. 6 10 The tsunami released = 4000 times as much energy as 22 1. 4 10 the earthquake in San Francisco.

Solving Exponentials and Logarithms Alg 2: 4. 5 • If the bases of two equal exponential functions are equal – the exponents are equal – If bases don’t look equal – try and make them (Ex 3) – Examples: 3 x = 32 7 x+2 = 72 x 48 x = 16 – Examples: • Logarithms are the same: common logarithms with common bases are equal – Examples: log 7(x+1) = log 75 – Examples: log 3(2 x+2) = log 33 x

Solving Exponentials - continued • Exponents without common bases – Get exponential by itself – Convert to logarithm • Examples: – 5 x = 7 3(2 x+1) = 15 6(x+1) + 3 = 12

Solving Logarithms – continued • Logarithms with logs only on one side – Use the properties of logarithms to solve – Get one log by itself – Must convert to exponential • Examples: (Using properties of logarithms) – Log 3(x – 5) = 2 – Log 2 x 2 = 8 log 45 x – log 3 = 1 log x + log (x+9) = 1

Exponential Inequalities • Set up equations the same but use inequality • Solve the same as equalities – Example:

Log Inequalities • Remember Domain of Log is • Only positive x values Examples:

Natural Logarithm • Inverse of natural base, e • Written as ln – Shorthand way to write loge – Properties are the same as for any other log • Examples: Inverse operations – ln e 3. 2 eln(x-5) e 2 ln x • Convert between e and ln – ex = 5 ln x = 43 ln e 2 x+ln ex

Base e and ln Equations • Examples: • Inequalities Ex:

Transforming Exponentials

Transforming Logarithms