OBJECTIVE 4 Eponential & Log Functions EXPONENTIAL FORM An eponential function is a function of the form where > 0 and. f ( ) SHAPE OF > increasing 0 < < decreasing
PROPERTIES OF THE BASIC EXPONENTIAL GRAPH Domain is (-, ) Range is (0, ) Horizontal Asymptote is y = 0 y-intercept is (0, ) DEFINITION OF e is an irrational numer (like π) e.7888845 The natural eponential function is f ( ) e GRAPHING EXPONENTIALS Graph. y =. y = - 3. y = + 4. y = + 5. y = -3 -
EXPONENTIAL APPLICATIONS Margaret Madison, DDS, estimates that her dental equipment loses one-sith of its value each year. a. Determine the value of an -ray machine after 5 years if it cost $6 thousand new.. Determine how long until the machine is worth less than $5 thousand. SOLVING AN EXPONENTIAL EQUATION Solve for :. 8 7. 8 3. 9 7 4. 5 5 LOGARITHMS The ase logarithm of, written log ( > 0 and ), is the eponent to which must e raised in order to yield. y = log is equivalent to y = 3
LOGARITHMS 3 = 8 so log 8 = 3 3 = 9 so log 3 9 = 5 3 = 5 so log 5 5 = 3 LOGARITHMS log 3 5 ecause 5 3 0 00 log 0 0.0 ecause 0 0. 0 log 9 3 ecause 9 9 3 f ( ) log a IS THE INVERSE FUNCTION OF f ( ) a Otain the graph of the inverse y reflecting across y = f ( ) a f ( ) log a 4
GRAPH PROPERTIES f ( ) a f ( ) log a Domain (, ) Domain( 0, ) Range ( 0, ) Range (, ) H. A. y 0 V. A. 0 ( 0,) (,0 ) GRAPHING LOGARITHMS Graph y = log Graph y = log ( +5) Graph y = log ( )+5 Graph y = 3log ( +) PROPERTIES log = log = 0 log = log = 5
LOGARITHMIC PROPERTIES log = 0 log = log 3 = 3 log 5 = 5 COMMON LOGARITHM The common logarithm is ase 0. We often write it omitting the ase: log = log 0 NATURAL LOGARITHM The natural logarithm is ase e. We write it as: ln = log e 6
SOLVING EQUATIONS WITH LOGARITHMS Solve Solve Solve Solve 4 (0 ) 66 log 4 3.7 6 4ln 3 PROPERTIES OF LOGARITHMS If, y, and > 0, then log ( y) log log log y log log log k k log y y EXPANDING LOGARITHMS Epand the epression. If possile, write your answer without eponents. y 3 ln log z y.. log y 3. 4. z log 5y 7
EXPANDING LOGARITHMS Epand the epression. If possile, write your answer without eponents. m ln pq ln n 3. 3. log 3 v v 3. 3 4. 7 3 4 ln 3 ( ) COMBINING LOGARITHMS Write the epression as a logarithm of a single epression.. log6 45 3log 6. log log3( 3) log3( 3 3 4) COMBINING LOGARITHMS Write the epression as a logarithm of a single epression.. log 7 log 6. log ( 5) log3( 3 5) 8
CHANGE OF BASE FORMULA Let, a, and e positive real numers. Then Eample: Evaluate log loga log a 0.77 log 6 STEPS FOR SOLVING EXPONENTIAL EQUATIONS. Isolate the power on one side of the equation.. Rewrite the equation in logarithmic form. 3. Evaluate the logarithm. 4. Solve for the variale. SOLVING EXPONENTIAL EQUATIONS Solve for. e 8. 3. 4. 00 5(0) 7 e e 5 3 30 3(0.75) 9 9
SOLVING EXPONENTIAL EQUATIONS Solve for.. 3. 7 3 7 4 80 5e 0.06 50 STEPS FOR SOLVING LOGARITHMIC EQUATIONS. Comine all logarithms using the properties of logarithms.. Isolate the logarithm on one side of the equation. 3. Rewrite the equation in eponential form. 4. Simplify and solve for the variale. 5. Check your answer!!! You may have answers that don t work. SOLVING LOGARITHMIC EQUATIONS Solve for.. 3. 4. 5ln 0 log 3 ( ) log 5 4 3log ln( 4) ln( ) ln(3 ) 0
SOLVING LOGARITHMIC EQUATIONS Solve for.. 3. 3 ln4 6.9 5. 4 log( 3 3) log log( 4) log log( 6) COMPOUND INTEREST The amount accumulated in an account earing interest compounded n times annually is nt r A ( t ) P n where P = principal invested r = interest rate (as a decimal) t = time in years COMPOUND INTEREST Suppose $5000 is invested at an interest rate of 8%. Find the amount in the account after ten years if the interest is compounded a. annually. semiannually c. daily
COMPOUND INTEREST What principal should e deposited at 8.375% compounded monthly to ensure the account will e worth $0,000 in 0 years? CONTINUOUSLY COMPOUNDED INTEREST The amount accumulated in an account earing interest compounded continuously is A( t) rt Pe where P = principal invested r = interest rate (as a decimal) t = time in years CONTINUOUSLY COMPOUNDED INTEREST $5000 is invested at an interest rate of 8%, compounded continuously. How much is in the account after 0 years?
CONTINUOUSLY COMPOUNDED INTEREST What principal should e deposited at 8.375% compounded continuously to ensure the account will e worth $0,000 in 0 years? EXPONENTIAL GROWTH AND DECAY Eponential growth of a population is given y the formula P 0 ( t) P e where P 0 = initial size of the population k = relative rate of growth (positive) or decay (negative) t = time kt EXPONENTIAL GROWTH The population of Phoeni, Arizona, in 000 was.3 million and growing continuously at a 3% rate. a. Assuming this trend continues, estimate the population of Phoeni in 00.. Determine graphically or numerically when this population might reach million. From Precalculus with Modeling and Visualization 3 rd ed. y Rockswold, 006, p.46, prolem 98. 3
EXPONENTIAL DECAY The radioactive element americium-4 has a half-life of 43 years and although etremely small amounts are used (aout 0.000 g), it is the most vital component of standard household smoke detectors. How many years will it take a 0-g mass of americium-4 to decay to.7 g? 4