Exponential and Logarithmic Functions

Exponential Functions Vocabulary Exponential Function Base (Common Ratio) Growth (Appreciation) Decay (Depreciation) Asymptote Inverse Function Logarithmic Function

Graphing Exponential Functions Make a Table of Values Enter values of X and solve for Y Plot on Graph Examples: Y = 2x Y = .5x

Appreciation 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 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: Inverse Operations If subtracting – add If adding – subtract If multiplying – divide If dividing – multiply x 1 2 3 4 5 y 6 9 12 15 18

Inverse Functions Steps for creating an inverse Example: f(x) = 2x – 3 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 Example: f(x) = 2x – 3

Logarithms Inverse of an exponential function Logbx = y b is the base (same as exponential function) Transfers 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

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 Product Property: logbx + logby = logb(x*y) Example: log48 + log432 Quotient Property: logbx – logby = logb(x/y) Example: log575 – log53 Power Property: logbxy = y*logbx Example: log285

More Logarithmic Properties Inverse Property: logbbx = x & blogbx = x Example: log775 Example: 10log 2 Change of Base: logbx = (logax ÷ logab) Example: log48 Example: log550

Natural Logarithm Inverse of natural base, e Written as ln Examples: Shorthand way to write loge Properties are the same as for any other log Examples: Convert between e and ln ex = 5 ln x = 43 ln e3.2 eln(x-5) e2ln x ln e2x+ln ex Used for constant (continuous) growth or decay Example: P 277 – Example 4 : Half life & decay

Solving Exponentials and Logarithms If bases of two equal exponential functions are equal – the exponents are equal bx = by if and only if x = y Examples: 3x = 32 7x+2 = 72x 48x = 162 Logarithms are the same: common logarithms with common bases are equal logbx = logby if and only if x = y Examples: log7(x+1) = log75 log3(2x+2) = log33x Logarithms with logs only on one side Use the properties of logarithms to solve Example: log3(x+7) = 3

Logarithmic Equations (Cont) Examples: (Using properties of logarithms) Log3(x – 5) = 2 log 45x – log 3 = 1 Log2x2 = 8 log x + log (x+9) = 1

Solving Logarithms - continued Exponents without common bases Use common log to set exponentials equal Use power property to bring down exponent Isolate the variable Divide out the logs – use the calculator Examples: 5x = 7 3(2x+1) = 15 6(x+1) + 3 = 12

Exponential Inequalities Set up equations the same but use inequality Solve the same as equalities Example: 2(n-1) > 2x106

Transforming Exponentials

Transforming Logarithms