Exponential and Logarithmic Equations. Solving Exponential Equations. Example. Solve = 2 8

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1 Eponential and Logarithmic Equations 1 Solving Eponential Equations Terms that have a base on one side and a power of that base on the other side can be solved using the property: m n a = a m= n Eample 1 Solve = = 4 = + + = 4 0 ( )( ) 1 = 0 = 1, = 1

2 Practice Solve 5 = = 5 + = + = 1 4 Solving Eponential Equations Watch for quadratic forms where we have b = ( b ) We can often factor epressions containing terms like b and b 5 Eample e Write in quadratic form Factor e 8= 0 ( ) e e 8= 0 ( e )( e + ) = 4 0 Use Zero Property e 4 = 0 or e + = 0 Solve for e e = 4 or e = Recall R(e ) >0 Take the ln of each side e = 4 = ln 4 6

3 Solving Eponential Equations General Guidelines Step 1: Isolate the eponential epression Put your eponential epression on one side everything outside of the eponential epression on the other side of your equation Step : Take the natural log of both sides The inverse operation of an eponential epression is a log. Make sure that you do the same thing to both sides of your equation to keep them equal to each other 7 Solving Eponential Equations Step : Use the properties of logs to pull the out of the eponent Step 4: Solve for Now that the variable is out of the eponent, solve for the variable using inverse operations to complete the problem 8 + Eample ( ) 1 Isolate the eponent Take the log of each side 510 = 1 + ( 10) 1 = = log 5 Solve for 1 = log 1 5 9

4 Practice e = 50 Take the ln of each side Solve for = ln50 = ln Solving Logarithmic Equations of the Form log b = y Step 1: Write as one log isolated on one side Get your log on one side everything outside of the log on the other side of your equation using inverse operations. Also use properties of logs to write it so that there is only one log Step : Use the definition of logarithms to write in eponential form A reminder that the definition of logarithms is the logarithmic function with base b, where b > 0 and b 0, and is defined as log b = y if and only if b y = 11 Solving Logarithmic Equations of the Form log b = y Step : Solve for Now that the variable is out of the log, solve for the variable using inverse operations to complete the problem Step 4: Verify the domain This is necessary as the domain of log() is strictly positive reals 1 4

5 Eample Use properties of log Rewrite as eponent Cross multiply Epand Solve and verify domain log ( + 4) log ( + ) = + 4 log = = = + 9 ( ) + 4 = = = 4 1 Eample + ln = 4 Subtract Property of log ln = 1 1 ln ( ) = 1 ln = Multiply by ( ) Write in eponential form = e Add Verify domain { e } = + e + e > Practice Solve for 1 log = log + 1 ( ) ( ) = ( )( + ) ( )( + ) 1= log - + log + 1 log10 = 1 log = = 5 5 1= 0 ( + 4)( ) = 0 = 4, = ( ) ( ) disgard negative solution and = 15 5

6 Solving Mied Equations Eponential or logarithmic equations that mi bases + 5 = 1 Equations that mi eponential and logarithmic epressions + ln = 5 Equations that mi transcendental and algebraic epressions + 5 = 5 Use Graphic Method to solve these mied equations 16 Eample + 5 = 1 17 Eample + ln =

7 Practice + 5 = 5 19 Remarks We often find it more useful to think in terms of the time required for money to double, instead of in interest rates and time compounded. It would be is simpler if you were told that anything invested will double in 7 years. 0 Eample Find time to double, if the interest rate is 6% compounded quarterly r P 1+ n nt.06 = P t ( ) 4 = P( 1.015) 4t t We want to know when = ( ) 4 4tt = ( ) 4 4t ln ( 1.015) = ln t ln = ln ln t = 4ln1.015 Answer: years 1 7

8 Eample Find time to double, if the interest rate is 6% compounded quarterly Answer: years Remarks We can approimate the time to double using 70 t = r% Eample: Find the approimate time to double, if the interest rate is 6% compounded quarterly 70 t = = years 6 Practice rt Pe 0.085t = Pe Find time to triple, if the interest rate is 8.5% compounded continuously 0.085t We want to know when e = 0.085t e = 0.085t = ln ln t = ln t = Answer: 1.9 years 4 8

9 Eample You invested in property near the River Walk in 004 for $5,000 and sold it in 008 for $685,000. What is the annual rate of return for this investment? r t = = 4 n =1 Using P 1+ n ( r) 4 $5, = 685, 000 ( 1+ r) 4 = 685,000 5, 000 ln 1+ = ln 65 ( ) 4 r = 65 nt 17 4ln( 1+ r) = ln ln ( 1+ r) = ln Eample 1 + r = e 1 17 ln ln 4 65 r = e 1 = You invested in property near the River Walk in 004 for $5,000 and sold it in 008 for $685,000. What is the annual rate of return for this investment? 1 17 ln ( 1+ r) = ln 4 65 Answer: 0.49% 6 Eample A model for the number of students at Palo Alto College that have heard the latest rumor might be.05d ( 1 ) N = P e where P is the total number of students at Palo Alto and d is the number of days that have elapsed since the rumor began.? If P = 8000 students, how many students will know the latest rumor in two days? In four days? 7 9

10 Two days Four days 8 10