Math- Lesson 8-7 Unit 5 (Part-) Notes 1) Solve Radical Equations ) Solve Eponential and Logarithmic Equations ) Check for Etraneous solutions 4) Find equations for graphs of eponential equations 5) Solve a) Cooling problems b) ph problems c) Sound Intensity Problems d) Money Problems e) Radioactive Decay Problems
Radical Equation: 1 +1 +1 Solving Equations Review Isolate the radical Undo the radical add 1 to both sides 4 square both sides 4 (6) 1 check 16 + + 18 16 1 6
Your turn: 5 Solve: 5 ( ) Isolate the radical Undo the radical square both sides 5 6 9 Get into standard form!!!!! 0 7, 9 14 0 ( 7)( ) (7) 5 (7) 16 4 () 5 () 1 1
4 4 4 The easiest eponential of all. Eponent = eponent log ( 4) log ( 7 4 4 7 9 0 6 5 0 ( 5)( 1) 5, 1 1 9) The easiest logarithm of all. Logarand = logarand Check for etraneous solutions
9 Solving using convert to same base 7 1 1 convert to same base * ( 1) Power of a power Eponent Property 4 4
Solving using log of a power property 9 ln 9 7 1 ln 7 1 ln 9 ( 1)ln 7 ln 9 ln 9 ln 7 ( 1) ln 9 ( 1)(1.5) 1.5 1.5-1.5-1.5 Take natural log of both side log of pwr property simplify simplify 0.5 1.5 * * 1 4 4
5 ln 5 ln Sometimes you can t rewrite the eponentials with the same bases so you have no choice. 5 7 1 ( ln 7 1 1)ln 7 ln 5 ln 5.4 1.1 +1.1 +1.1 1.1. 4 - - ( 1) ln 7 ln 5 ( 1)(1.1) 1.1 1.4 1.4 1.4 0.85
Solve using undo the eponential 5 1 1 7-5 -5 Isolate the eponential Undo the eponential 1 log 1 1 ln ln 0.609 +1 +1 Change of base formula 1.609 0.815
Some functions don t have domain of all real numbers equations of these types may have etraneous solutions Etraneous solution: an apparent solution that does not work when plugged back into the original equation. You MUST check the solutions in the original equation for any equation this is of the function type that has a restricted domain. Square root equations Log equations Rational equations
Solving Logarithmic Equations log 5 5 Power property of logarithms log 5 5 ln 5 ln 5 5 ln ln 5 Change of base Use inverse property of multiplication 5(0.407). 154
Solving Logs requiring condensing the product. log log( 5) log ( 5) Isolate the logarithm undo the logarithm condense the product 10 10 Quadratic put in standard form 10 100 5 50 0 ( 10)( 5) 0 0 factor Divide both sides by = 10, -5 Zero factor property
Check the solution: log = 10, -5 log( 5) log( *10) log(105) log( 0) log(5) Condense the product log( 0*5) log 100 Convert to eponent form 10 100 Checks
Check the solution: log = 10, -5 log( 5) log( ( 5)) log( 5 5) -5 is NOT a solution log( 10) log( 10) log(10) 10 10 This doesn t make sense punch this into your calculator. Or, convert to eponent form There is NO eponent that will cause a positive number to equal a negative number.
More complicated Logarithmic Equations log 5 7 - - log 5 5 ( )log 5 5 ln 5 ( ) ln 5 ln 5 ln 5 ln 5 ln 5 Isolate the logarithm undo the logarithm Power property of logarithms Change of base Use inverse property of multiplication Add to both sides. 5(0.407) 4. 154
More complicated Logarithmic Equations log 5 7 - - log 5 5 Isolate the logarithm undo the logarithm 5 ( ) 5 Solve by graphing?
Solving Logarithmic Equations log (5 4 1) Isolate the logarithm convert to eponential 4 5 1 Inverse of log base 4 is eponent base 4. 5-1 = 64 5 65 = 1 log 4 (5*11) Subtract 1 from both sides Divide both sides by 5 Plug back in to check! log 4 64 Checks
We can rewrite the base of any eponential as a power of e. y y k e k y e k e k ln y e 0.69 k e 0.69 0.69
y 4 y 1. 1 y 1. 01 y 0. 85 y 0. 5 How can you tell if a base B eponential is growth or decay? y B Growth: B > 1 Decay: 0 < B < 1
Rewrite the following eponential equations as base e eponentials. y 4 y 1. 1 y 1. 01 y 0. 85 e 1. 86 e 0. 095 e 0. 010 e 0. 16 y 0. 5 e 1. 86
Find the equation of the graph. 1. Horizontal asymptote: y = 5 y AB 5. Passes through: (, y) = (0, 8) A 8 AB 0 5 8 A 5 y B 5. Passes through: (, y) = (1, 7) 7 B 1 5 B B y AB Convert to base e. y e 0.4055 k y 5 k e ln k 5 k 0.4055
Find the equation of the graph. 1. Find for the point (, 5.5) 5 0.4055 y y e 5 5 5.5 5.5 5 5.5 e 0.4055 5.5 5 e 0. 4055 5 0.1667 log 0.1667 4.4 0.1667 e 0. 4055 ln 0.1667 0.4055 4.4
A cup of hot water is taken out of the microwave oven. Its initial temperature is 100 C. It is placed on the counter in a room whose temperature is 0 C. In 5 minutes it has cooled to 7 C. When will it reach 40 C. 1. Draw a graph that shows temperature as a function of time.. What is the equation of the graph? (use the following equation). T ( t) AB t T ( t) 70(0.9) t 0. T(t) = 40 C. Substitute and solve for t. k t 40 70(0.9) 0 t 17.7 min
y AB Growth (decay) factor Initial Value Initial Value does not mean y-intercept if the graph has been shifted up/down. y Ae k m Growth (decay) rate Horizontal asymptote
Loudness L( I) I 10log 10 1 The loudness of an ambulance was measured to be 10 db. What is the sound intensity? (in w/m^) I 10 10log 10 I 1 log 1 10 1 I 10 1 10 1 I 10 1 I 10 0 1 *10 1 Properties of eponents!!!
Acidity ph = - log [H + ] The ph of baking soda is 8.6. What is the hydrogen ion concentration? 8.6 = - log [] 10 8. 6 mole/li 9.510 mole/li
A bank compounds interest continuously. The annual interest rate is 5.5%. How long would it take for the money in the account to triple? rt A( t) A e 0 A 0 A e 0 e 0. 055t 0.055t ln 0.055t t 19.97 yrs
The half life of Carbon-14 (a radioactive isotope of carbon), is 570 years. Calculate the decay rate for carbon-14. The decay rated is the k of the eponent of e. A( t) A0e 0.5A 0.5 ln 0 0.5 e A e 0 kt 570( k ) k (570) 570k k 0. 0001 yr If there were 5 grams of C-14 initially, how many grams will be present after 15,000 years? A(15,000) A( 15,000) 5e 0.0001(15,000) 0.8 gms If the amount of C-14 was 5% of the original amount, how old is the specimen? 0.05A t 0 ln 0.05 A e 0 4,964 yrs 0.0001( t) 0.0001t