Exponential and Logarithmic 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 = 2x state D: and R: Y = (1\2)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:

Switch the x- and y-values in each ordered pair. • • • • • 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 2 Graph each ordered pair and connect them. Switch the x- and y-values in each ordered pair. • • x 1 2 3 5 y 4 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 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. The reverse order of operations: Addition or Subtraction Multiplication or Division Exponents Parentheses Helpful Hint

Inverse Functions - Examples Example: f(x) = 2x – 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. Charge c is a function of list price L. c = 0.80L + 2.50

c – 2.50 = L 0.80 Example Continued Step 2 Find the inverse function that models list price as a function of the change. Subtract 2.50 from both sides. c – 2.50 = 0.80L 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 for c. L = 13.70 – 2.50 0.80 = 14 The list price of the CD is $14. Check c = 0.80L + 2.50 = 0.80(14) + 2.50 Substitute. = 11.20 + 2.50 = 13.70 

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. Read logb a= x, as “the log base b of a is x.” Notice that the log is the exponent. Reading Math

Example 1: Converting from Exponential to Logarithmic Form Write each exponential equation in logarithmic form. Exponential Equation Logarithmic Form 35 = 243 25 = 5 104 = 10,000 6–1 = ab = c The base of the exponent becomes the base of the logarithm. log3243 = 5 1 2 1 2 log255 = The exponent is the logarithm. log1010,000 = 4 1 6 log6 = –1 1 6 An exponent (or log) can be negative. logac =b 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 log99 = 1 log2512 = 9 log82 = log4 = –2 logb1 = 0 The base of the logarithm becomes the base of the power. 91 = 9 The logarithm is the exponent. 29 = 512 1 3 1 3 8 = 2 A logarithm can be a negative number. 1 1 16 4–2 = 16 Any nonzero base to the zero power is 1. b0 = 1

Solving & Graphing Logarithms Write out in exponential form: b? = x What value needs to go in for ? Example: log327 = ? 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 2x and then log2x

Properties of Logarithms Alg 2: 4.4 Product Property: logbx + logby = logb(x*y) Example: log64 + log69 Quotient Property: logbx – logby = logb(x/y) Example: log5100 – log54 Power Property: logbxy = y*logbx Examples: log 104 log5252

Example 1: Product Property Express log64 + log69 as a single logarithm. Simplify. log64 + log69 To add the logarithms, multiply the numbers. log6 (4  9) log6 36 Simplify. 2 Think: 6? = 36.

Example 2: Quotient Property Express log5100 – log54 as a single logarithm. Simplify, if possible. log5100 – log5 4 To subtract the logarithms, divide the numbers. log5(100 ÷ 4) log525 Simplify. 2 Think: 5? = 25.

Example 3 – Power Property Express as a product. Simplify, if possibly. a. log104 b. log5252 4log10 2log525 Because 101 = 10, log 10 = 1. Because 52 = 25, log525 = 2. 4(1) = 4 2(2) = 4

More Logarithmic Properties Inverse Property: logbbx = x & blogbx = x Example: log775 Example: 10log 2 Change of Base: logbx = Example: log48 Example: log550 Change of base in Calculator: MATH, logBASE( 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 in1989? 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 13.95 = log 10 E æ ç è ö ÷ ø Multiply both sides by . 3 2 11.8 13.95 = log 10 E æ ç è ö ÷ 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.

Apply the Quotient Property of Logarithms. Example Continued Substitute 6.9 for M. Multiply both sides by . 3 2 Simplify. Apply the Quotient Property of Logarithms.

Given the definition of a logarithm, the logarithm is the exponent. 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. The tsunami released = 4000 times as much energy as the earthquake in San Francisco. 1.4  1022 5.6  1025

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: 3x = 32 7x+2 = 72x 48x = 16 Examples: Logarithms are the same: common logarithms with common bases are equal Examples: log7(x+1) = log75 log3(2x+2) = log33x

Solving Exponentials - continued Exponents without common bases Get exponential by itself Convert to logarithm Examples: 5x = 7 3(2x+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) Log3(x – 5) = 2 log 45x – log 3 = 1 Log2x2 = 8 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 e3.2 eln(x-5) e2ln x ln e2x+ln ex Convert between e and ln ex = 5 ln x = 43

Base e and ln Equations Examples: Inequalities Ex:

Transforming Exponentials

Transforming Logarithms