Substitute 4 for x in the function, Simplify.

Transcription

1 Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The domain of the function is all real numers and the range is all positive real numers. Note: In the eponential function the variale is an eponent. If the ase is e =, then ( ) Evaluating an Eponential Function Eample: f ( ) + = Find f ( 4) without using a calculator. f = e. This is called the natural eponential function. Identification/Analysis This is an eponential function with ase. 4+ f ( 4) = Sustitute 4 for in the function, Simplify. = 1 1 n 1 = = Recall the definition of a negative eponent is a =. 9 a Eample: Evaluate ( ) 4 f = at =, using a calculator. Round your answer to the nearest thousandth. Identification/Analysis This is an eponential function with ase 4. Find the appropriate calculator key. Calculators use one of two keys. Caret key type: 4 ^ [ENTER] [ ] Power key type: 4 [ y ] [ = ] f = 4 = 64 ( ) (Note: Square rackets indicate a calculator key.) [ ^] Caret or power operator key y or y ase raised to a power key. Check your calculator manual if you do not otain 64.

2 Page of 19 4 Eample: Evaluate f ( ) e = at =, using a calculator. Round your answer to the nearest thousandth. Identification/Analysis This is an eponential function with ase e, called the natural eponential function. ( ) 4 f ( ) = e Sustitute for. Simplify. = e For most calculators types: Find the e key on your calculator. [] e [ENTER] Enter the sign e followed y. f ( ) = e 7.89 Check your calculator manual if needed to determine the evaluation sequence. Logarithms A logarithm is an eponent. It is the eponent to which the ase must e raised to produce a given numer. For eample, since = 8, then is called the logarithm with ase. It is written as: log 8 = is the eponent to which must e raised to produce 8. = 8 is called the eponential form. log 8 = is called the logarithmic form. General definition of the logarithm function with ase, > 0 and 1, > 0 is given y: log = y is equivalent to y = For eample: log 9 = is equivalent to = 9 The function is written as f ( ) = log and is read as log ase of or logarithm of with ase.

3 Page of 19 The two most often used ases are 10 and e. These are called the common logarithm and natural logarithm. Common logarithm A common logarithm is a logarithm with a ase 10. When the ase is not indicated, such as f ( ) = log, a ase 10 is implied. y = log is equivalent to 10 y = Natural logarithm A natural logarithm is a logarithm with ase e. f =. It is written as ( ) ln y = ln is equivalent to y e = Eample: Write log515 = in eponential form. Identification/Analysis Use the definition of a logarithm to write the logarithm in eponential form. The ase on the logarithm is 5 so the ase for eponential form is also 5. log515 = The logarithm is ase 5, so the ase of the eponential is also 5. 5 = 15 The logarithm is equal to, so the eponent will e Eample: Write log 0.01 = in eponential form. Identification/Analysis Use the definition of a logarithm to write the logarithm in eponential form. The ase on the logarithm is not indicated so ase 10 is implied. log 0.01 = The logarithm is ase 10, so the ase of the eponential is 10. The logarithm is equal to, so the eponent will e 10 = 0.01

4 Page 4 of 19 Eample: Write 7 = 4 in logarithmic form. Identification/Analysis Use the definition of a logarithm to write the eponential as a logarithm. The ase of the eponential is 7, so the ase of the logarithm is also 7. 7 = 4 The eponential is ase 7, so the ase of the logarithm is also 7. log 4 = The eponent is, so the logarithm will e equal to. 7 Eample: Write 10 = in logarithmic form. Identification/Analysis Write the radical as an eponent. Use the definition of a logarithm to write in logarithmic form. 10 = Rewrite the radical as an eponent. ( 10) 1/ = The ase of the eponential is 10 so the ase of the logarithm is 1 log = also 10. The ase of the common logarithm is also 10. The eponent is1/, so the logarithm will e equal to 1/. Eample: Evaluate log 81 Identification/Analysis Use the definition of a logarithm to write the epression in eponential form. The ase on the logarithm is, so the ase for eponential form is also. log81 = y Set the logarithm equal to y. y = 81 Rewrite the logarithm in eponential form. The ase is and the eponent is y. 4 y = 81= ***= Determine an appropriate eponent. (Factor 81 or write 81 as a power of.) log81 = 4 The logarithm is the eponent so the logarithm equals 4

5 Page 5 of 19 Eample: Evaluate log1000 Identification/Analysis Use the definition of a logarithm to write the epression in eponential form. The ase on the logarithm is not indicated so the ase is 10. log1000 = y Set the logarithm equal to y. 10 = 1000 Rewrite the logarithm in eponential form. 10 = 10 Factor1000 or write 1000 as a power of 10 to determine the eponent. log1000 = The logarithm is the eponent so the logarithm equals. Eample: Evaluate log ( 0.01 ) Identification/Analysis Use the definition of a logarithm to write the epression in eponential form. The ase on the logarithm is not indicated so the ase is 10. log 0.01 = y Set the logarithm equal to y. 10 = 0.01 Rewrite the logarithm in eponential form. 1 Write 0.01 as a fraction. 10 = 100 y 1 Write 100 as a power of 10 to help determine the eponent. 10 = = log 0.01 = The logarithm is the eponent so the logarithm equals. Eample: Evaluate 5 ln e Identification/Analysis This is a natural logarithm with ase e. Use the definition of a logarithm to write the epression in eponential form with ase e. 5 ln e = y Set the logarithm equal to y. y 5 e = e Rewrite the logarithm in eponential form. 5 ln e = 5 The logarithm is the eponent so the logarithm equals.

6 Page 6 of 19 Properties of Logarithm Functions The first column lists a property of logarithms. The second column provides a rational for the property ased on the definition of a logarithm. While the third column provides an eample of the property. Property Rational Eample For > 0, 1 log 1 0 = 1 0 log1 = 0 Since 10 = 1 = 0 1 log = 1 = log = 1 Since 1 = log = = log4 4 = Since 4 = 4 log = y = is equivalent to y = log Sustitute y = log into the first statement to validate the identity. ln e = ( ) log 5 10 = 5 If log = log y then = y For > 0 and y > 0. The logarithms have the same ase and are equal so the arguments are equal. If log ( + 7) = log ( ) then ( 7) + = Solve for. 7 = Eample: Evaluate log 81 Identification/Analysis This is a logarithm with ase, so the ase of the eponential will also e ase. 4 log81 = log Write 81 as a power of three since the ase on the logarithm is. 4 log = 4 Evaluate the logarithm using log =.

7 Page 7 of 19 Eample: Evaluate ln e Identification/Analysis This is the natural logarithm with ase e, so the ase of the eponential will also e e. 1/ ln e = ln e Write the radical in eponential form. 1/ 1 Evaluate the logarithm since the ase of the logarithm and the ln e = eponential are oth e. Eample: Evaluate log Identification/Analysis The ase on the logarithm is not indicated so the ase is Write as a fraction. log = log = log10 4 Write the fraction as a power of 10 since the ase is 10 log = 4 Evaluate the logarithm since the ases of oth are 10. Eample: Evaluate: 10 ( + ) log 5 Identification/Analysis This is an eponential with ase 10. In the eponent is a common logarithm with ase 10. log( 5+ ) 10 = ( 5 + ) Since the ase of the eponential and the ase of the logarithm log are the same, apply =.

8 Page 8 of 19 Additional Properties of Logarithms Listed elow are properties of logarithms used to epand and condense logarithmic epressions. The properties apply to any ase ut most applications use common ( log ) or natural logarithms ( ln ). Name of the Property General Rule Assume > 0, 1, m > 0 and n > 0 Eample Assume > 0 and y > 0 Stated in Words Product Rule log ( mn) = log m + log n ( ) log = log + log The logarithm of a product equals the logarithm of the first factor plus the logarithm of the second factor. Quotient Rule m log log m logn n = ln ln ln y y = The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator. Power Rule n log m = nlog m log = log The logarithm of a power equals the eponent times the logarithm. Change of Base Formula log log log m m = or log ln m m = ln Note: The quotient can e written with any ase as long as the ase in numerator and denominator are the same. log 5 log 5 = log The logarithm ase of m equals the common logarithm of m divided y the common logarithm of the original ase For the eamples elow, only common and natural logarithms are used even though the properties apply to any ase. Assume are variales used are positive numers.

9 Page 9 of 19 Eample: Epand the logarithm ln ( ) Identification/Analysis Epand means to write the logarithm as a sum or difference of simpler logarithms. This is a natural logarithm of a product. ln ( ) = ln + ln Apply the product rule. The log of a product is the sum of the logs. Eample: Epand. log a Identification/Analysis Epand means to write the logarithm as a sum or difference of simpler logarithms. This common logarithm contains a quotient and a power. log a Apply the quotient rule. The log of a quotient is the difference of = log a log the logs. = loga log Apply the power rule. The log of an eponential is the eponent times the log. Eample: Epand. ln y Identification/Analysis Epand means to write the logarithm as a sum or difference of simpler logarithms. This natural logarithm contains a quotient and a product. ln Apply the quotient rule. The log of a quotient is the difference of = ln ln ( y) y the logs. = ln ( ln+ lny) Apply the product rule. The log of a product is the sum of logs. Be sure to put the sum in parentheses. = ln ln ln y Distriute the negative. (This is not a required step.)

10 Page 10 of 19 Eample: Epand the logarithm and simplify. 100y log Identification/Analysis Epand means to write the logarithm as a sum or difference of simpler logarithms. This is a common logarithm contains a quotient, product and a power. 100 log y Apply the quotient rule. = log ( 100y) log = log100 + log y log Apply the product rule. 1/ = log10 + log y log Write 100 as a power of 10 since the ase on the logarithm is 10. Write the radical as an eponent. 1 = + logy log Apply log = since the ases are oth 10. Apply the power rule. The following eamples use the properties of logarithms in the opposite manner. In other words you start with an epression containing sums and difference of logarithms and condense to a single logarithm Eample: Condense. Write as a single logarithm. log + log y log z Identification/Analysis Condense means to write as a single logarithm. There is a sum and difference of common logarithms. In general when there are several terms, apply the rules one at a time moving from left to right. Since there is a sum and difference two rules apply. log + log y log z = log ( y) log z Apply the product rule in the opposite manner. The sum of logs equals the log of a product. log y Apply the quotient rule. The difference of logs equals the log of = z a quotient. The original is written as a single logarithm.

11 Page 11 of 19 Eample: Condense. Write as a single logarithm. 1 ln ln16 Identification/Analysis Condense means to write as a single logarithm. There is a difference of logarithms and a power times a logarithm. 1 1/ Apply the power rule. ln ln16 = ln ln16 Apply the quotient rule. = ln 16 1/ = ln 4 1 log log log Eample: Condense. ( y+ z) 1/ Simplify. 16 = 16 The original is written as a single logarithm. Identification/Analysis Condense means to write as a single logarithm. There is a sum and difference of logarithms. There are also parenthesis and a power times a logarithm. 1 Since there is parenthesis, first condense within the parenthesis. ( log log y+ log z) 1 Apply the power rule. = ( log log y + log z) 1 In general apply the rules one at a time from left to right. = log + log z y Apply the quotient rule. 1 z Apply the product rule. = log y 1/ log z = or log z Apply the power rule. Leave as an eponent or write as a radical. y y The original is written as a single logarithm.

12 Page 1 of 19 Eample: Condense 4 log log y log z. Identification/Analysis Condense means to write as a single logarithm. There are two differences of logarithms and a logarithm times a power. 4 log log y log z Apply the power rule. 4 = log log y log z 4 There are three terms apply the rules from left to right. = log log z y Apply the quotient rule. 4 log Apply the quotient rule. = yz The original is written as a single logarithm. Eample: Use a calculator to evaluate log7. Round to the nearest thousandth. Identification/Analysis The logarithm is ase 7. To evaluate with the calculator it must e written with a ase of 10 or e. Use the change of ase formula. log ln Apply the change of ase formula. log7 = or log7 = log 7 ln 7 Use either the common log or natural log. log Type log7 = [ log] [ ] [ log] 7 [ ENTER] and round appropriately. log 7

13 Page 1 of 19 Domain of the logarithmic function The function f( ) = log, > 0, 1 has a domain > 0 Also written {, > 0} or using interval notation ( 0, ) In words the argument of the logarithm must e positive. The ase of the logarithm does not affect the domain. Eample: Find the domain of f( ) = log( 7) Identification/Analysis The argument of the logarithm has to e positive. The ase of the logarithm does not affect the domain. 7 > 0 The argument of a logarithm must e positive. ( ) > 7 Solve for., > 7 7, State the domain using set uilder or interval notation. { }. or ( ) Eample: Find the domain of f ( ) = ln( ) Identification/Analysis The argument of the logarithm has to e positive. The ase of the logarithm does not affect the domain. ( ) > 0 The argument of a logarithm must e positive. > Solve for. Rememer when you divide oth sides y a negative numer the < direction of the inequality is reversed. State the domain using set uilder or interval notation., < or,

14 Page 14 of 19 Solving Eponential and Logarithmic Equations An eponential equation is an equation where the variale is an eponent. In general, consolidate the eponential terms, then isolate the eponential term containing the variale. Take the logarithm of oth sides of the equation. Simplify the logarithm or use the power rule to write the variale as a factor instead of an eponent. Eample: Solve e + 4 = 7 Identification/Analysis This is an equation where the variale is an eponent. The ase of the eponential is e so use the natural logarithm. e + 4= 7 Isolate the term containing the variale. e = ln e = ln Take the natural logarithm of oth sides. = ln Simplify the natural logarithm of e to a power Solve for. Check/verification ln e + 4= 7 Sustitute = ln in the original equation. Simplify. + 4 = 7 True, so = ln is a solution. Eample: Solve 10 + = 15 Solve eactly then use your calculator to estimate the solution to the nearest thousandth. Identification/Analysis This is an equation where the variale is an eponent. The ase of the eponential is 10 so use common logarithms. 10 = 15 The term with the variale is isolated. log10 = log15 Take the common logarithm of oth sides. + = log15 Simplify the common logarithm of 10 to a power. = log Solve eactly for. Approimate with your calculator. Check/verification ( log15 ) 10 = 15 Sustitute = log15 into the original equation. log15 10 = = 15 Simplify the right side. Comine +. The ase of the eponential and the logarithm are the same. True, so = log15 is a solution.

15 Page 15 of 19 Eample: Solve 1 + = 7 Identification/Analysis This is an equation where the variale is an eponent. There are two distinct ases, and 7. Both ases can e written using ase. 1 = 7 Both eponentials can e written using ase. 1 = Write each eponential with ase. + 1= Since the ases are equal the eponents are equal. = Solve for. = 1 Check/verification 1 ( ) 1 = 7 Sustitute = 1 into the original equation. = 7 Simplify. True so, = 1 is a solution. Eample: Solve + 1 = Find the eact Identification/Analysis This is an equation where the variale is an eponent. There are two distinct ases, and. The ases do not have a common factor so we cannot write them using the same ase + 1 = The ases are distinct and cannot e written using the same ase. log ( 1) = log + Take the log of oth sides. Use either common or natural log. log = + 1 log Apply the power rule. Solve for. Check/verification = ( ) log = log + log Distriute log. log log = log Collect the variale on one side of the equation. ( log log ) = log Factor out the variale. log = log log log = log log so Divide oth sides y ( log log ), the coefficient of. Approimate with a calculator. Round to thousandths. Sustitute the approimated value for in the original equation. Evaluate with a calculator. The two sides are.

16 Page 16 of 19 Logarithmic Equations A logarithmic equation is an equation that has a logarithmic epression containing a variale. Eample: Solve log( 5) + 6 = 8 Identification/Analysis This is a logarithmic equation with a common logarithm. There is one logarithmic term. log( 5) + 6 = 8 There is a logarithmic term which contains a variale. Check/verification ( ) log ( 5) = Isolate the logarithmic term. 10 ( 5) = Use the definition of a logarithm to write in eponential form = Solve for. = 105 Simplify. log = 8 Sustitute = 105 in the original equation. log = 8 Simplify each side independently. + 6= 8 Use 100 = 10 to simplify the common logarithm. 8= 8 True so = 105 is a solution. Eample: Solve ( ) + ln = 5 Solve eactly then use your calculator to estimate the solution to the nearest thousandth. Identification/Analysis This is a logarithmic equation with a natural logarithm. There is one logarithmic term. + ln = 5 There is a logarithmic term which contains a variale. ( ) Check/verification ( ) ln ( ) = Isolate the logarithmic term. e ( ) = Use the definition of a logarithm to write in eponential form. e ln 9.89 = 5 Sustitute = 9.89 in the original equation. = + Solve for. Approimate with a calculator Simplify the left side using a calculator. The two sides are approimately equal.

17 Page 17 of 19 Eample: Solve log log 4 = Identification/Analysis This is a logarithmic equation with a common logarithm. There are two logarithmic terms and a term without a logarithm. Use properties of logarithms to condense to a single logarithmic term. log log 4 = There are two logarithmic terms. log = Apply the quotient rule. 4 Use the definition of a logarithm to write in eponential form. 10 = 4 = 4*10 = 400 Solve for and simplify. Check/Verification log 400 log 4 = Sustitute = 400 in the original equation. = Evaluate the left side using a calculator. True so = 400 is a solution. Eample: Solve ln ( + 1) = ln 7 Identification/Analysis This is a logarithmic equation with a natural logarithm. There are two logarithmic terms and no terms without a logarithm. ln + 1 = ln 7 There is a single logarithm on each side of the equation. ( ) ( + 1) = 7 The logarithms are equal so the arguments are equal. = 6 = Solve for. Check/Verification ln ( * + 1) = ln 7 Sustitute = in the original equation. ln 7 = ln 7 Simplify. True so =.

18 Page 18 of 19 Eample: Solve ( ) log log = Solve eactly then use a calculator to approimate the solution to the nearest thousandth. Identification/Analysis This is a logarithmic equation with a common logarithm. There are two logarithmic terms. Use properties of logarithms to condense to a single logarithmic term. log log = There are two logarithmic terms. ( ) ( )( ) log + 1 = Apply the product rule. ( )( ) 10 = + 1 Use the definition of a logarithm to write in eponential form. 100 = + 1 Solve for = 0 Second degree equation = 0 Factor or use the quadratic equation. ( )( ) Check/Verification ( ) = 5 or = 4 log log 5 = Sustitute = 5 in the original equation. Notice this requires taking the logarithm of a negative numer. The domain of the logarithm requires the argument to e positive so = 5 is not a solution to the equation. + + = Sustitute = 4 in the original equation. log ( 1) log 4 = Evaluate the left side using a calculator. True so = 4 is a solution.

19 Page 19 of 19 Eample: Solve ( ) ( ) log + = log 1 + log Identification/Analysis This is a logarithmic equation with a common logarithm. There are two logarithmic terms and no terms without a logarithm log + = log 1 + log There are three logarithmic terms. Condense each side. ( ) ( ) ( + ) = ( ) ( ) ( 1 )* log log 1 * Apply the product rule. + = The logarithms are equal so the arguments are equal. + = Solve for. 6= = = in the original equation. Check/Verification log ( + ) = log ( 1) + log Sustitute log ( 6) = log ( ) + log Simplify. log 6 = log ( *) Apply the product rule to the right side = Or approimate each side using a calculator. In either case the statement is true so = is a solution.