Ch+8+0910

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 * __**Roles of Wiki**__ || __**Student **__ ||
 * ** Editor ** || ** Christian M. ** ||
 * ** Multimedia ** || ** Evan N. ** ||
 * ** Text ** || ** Joe P. and Leann M. ** ||
 * ** Practice Problems ** || ** Alisha Q ** ||
 * ** Career/Real Life App ** || ** Rachel R. ** ||

= = =HOW ARE LOGARITHMS RELATED TO REAL LIFE??? =\= =media type="custom" key="5815505"=

** VOCABULARY ** the viewed value || || same in all logarithmic expressions and b can not 1 || || written as just "log" ||  || exponential expression ||  || the input values with the output. ||  || value ||   || and the inverse is y = b ||  || as ln ||  || = = == =** __8.1 Exponential Growth__ (pg. 465 - 473) To graph an exponetial fuction you will need to make a table of values. There will be y=a2^x graphs and y=ab^x. In both of these a>0 and b>1. In a graph of y=ab^(x-h)+k, h will be the horizontal shift and k will be the vertical shift. Also you may need to find growth factor by using the function y=a(1+r)^t, where a is initial amount and r is the percent increase and t is time. **= = = = = Real Life application- Exponential Growth can be used in real life situations like population growth over time. Or a banking account that shows amount of interest over a specific number of year
 * ~ # ||~ Word ||~ Definition ||  ||
 * ~ 1 || asymptote || a line that never reaches
 * ~ 2 || base || the value that has to be the
 * ~ 3 || common logarithm || it is a log with a base of 10; sometimes
 * ~ 4 || exponential equation || an equation that contains more than one
 * ~ 5 || inverse function || the function that is the result of switching
 * ~ 6 || logarithmic equation || an equation that has a logarithm of a
 * ~ 7 || logarithmic function || a function of y = logb x, where b > 1;
 * ~ 8 || natural logarithm || a logarithm that has a base of // e, // written

Practice Problems:
Identify y-intercept & asymptote of the graph of the function. 1. y=5^x 2. y= 3 **·** 2^x-1 Graph the function, state the domain and range 1. y= 4 **·** 5^x-1 3. y= 4 **·** 2^x-3 +1

To graph exponential decay you will also use a table of values. you will use the same type of functions but most likely the b will be a fraction. the shifts are the same but the decay factor is not; the fuction for that is y=a(1-r)^t but a, r and t are the same thing. Real Life Application- Exponential Decay can be used in science. It can be used to find the half life of compounds and radioactive elements
 * __8.2 Exponential Decay__ (pg. 474 - 479)**

**Practice Problems:**  Tell whether the function is growth or decay. 1. f(x)= 8 **·** 7^-x 2. f(x)= 5(1/8)^-x Graph and state domain and range 1. y= (1/3)^x-2 2. y= (1/3)^x -2

"e" is a irrational, non-terminating, non-repeating decimal (like pi) that rounded equals 2.72. It is used in natural logarithms, which is like a common log but instead of base 10 the base is "e". to simplify use the properties of logs.
 * __8.3 The Number e__ (pg. 480 - 485)**

Continuously Compounded Interest Formula: A=Pe^rt Real Life Application- e can be used in finding compound interest. The more often the person compounds their money (monthly, weekly, daily, hourly,) the closer their interest rate gets to approaching e.



**Practice Problems:** Simplify. 1. e^-2 **·** 3e^7 2. (4e^-2)^3 Evaulate with calculator. 1. e^1.7 2. -4e^-3

First see definition at top of page. these functions are the same as an exponential equation. This is a log: logbaseb a=x this is a exponential b^x= a the variables are just in different places. to solve you either have to change log to exponetial or use this formula loga/logb when adding log functions you want to multiply and when subtracting you want to divide.
 * __8.4 Logarithmic Functions__ (pg. 486 - 492)**

**Practice problems: ** Evaluate without using a calculator. 1. log7 (343) 2. log4(4^-0.38) Using inverses, simplify the expression. 1. 35^log35(x) Find the inverse. 1. y= ln 6x

Logs have properties similar to exponents. You can see these properties below. Also these properties will help you condense and expand logs. x = 2 // || x - loga y ||= log (15/5) = log 15 - log 5\ x = 0.47712 || || // x // =// 10 // log // 4 // ( // 64 // )x = 256 ||
 * __8.5 Properties of Logarithms__ (pg. 493 - 500)**
 * ~ ** PROPERTIES ** ||~ ** FORMULAS ** ||~ ** EXAMPLES ** ||
 * = ** Product Property ** ||= // logb(x · y) = logb(x) + logb(y). // ||= // log10(5 · 20) = log10(5) + log10(20)
 * = ** Quotient Property ** ||= loga (x/y) = log
 * = ** Power Property ** ||= // x // =// b // log // b // ( // x // ). =

Use change of base formula to evaluate. 1. log 7 (12) 2. log 9 (5/16)
 * ~ Change of Bases ||~  ||
 * ** Formula ** || [[image:math_formula2.jpg]] ||
 * ** Example ** || [[image:change1.jpg]] ||
 * Practice Problems: **


 * __8.6 S​olving Exponential and Logarithmic Equations__ (pg. 501 - 508)**

When trying to solve exponential equations, there is a simpler way. Try getting the bases to the same number. (bx = bx)

Ex:  ** Solve 101– // x // = 104 ** 1 – // x // = 4 1 – 4 = // x // ** –3 = // x //**

When trying to solve logarithmic equations, there is a simpler way. ​Try getting the bases to the same number. (log b x= logbx)

Ex:  ** Solve // log // b( // x // 2) = // log // b(2 // x // – 1). ** // x // <span style="color: #2ae01f; font-family: 'Comic Sans MS',cursive;">2 = 2 <span style="font-family: 'Comic Sans MS',cursive;">// x // <span style="color: #2ae01f; font-family: 'Comic Sans MS',cursive;"> – 1 <span style="font-family: 'Comic Sans MS',cursive;"> // x // <span style="color: #2ae01f; font-family: 'Comic Sans MS',cursive;">2 – 2 <span style="font-family: 'Comic Sans MS',cursive;">// x // <span style="color: #2ae01f; font-family: 'Comic Sans MS',cursive;"> + 1 = 0 ( <span style="font-family: 'Comic Sans MS',cursive;">// x // <span style="color: #2ae01f; font-family: 'Comic Sans MS',cursive;"> – 1)( <span style="font-family: 'Comic Sans MS',cursive;">// x // <span style="color: #2ae01f; font-family: 'Comic Sans MS',cursive;"> – 1) = 0
 * // x // = 1 **

<span style="color: #0000ff; font-family: 'Comic Sans MS',cursive;">** Practice Problems: Solve. 1. 10^x-3 = 100^4x-5 2. 10^2x + 3 = 8 3. 1/4(4)^2x + 1 = 5

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif;">__8.7 Modeling__ (pg. 509 - 516) **

9 = ab2 9/b2 =a 20.25 = 9/b2 (b4) 20.25= 9b2 a = 4 b = 1.5 y = 4 (a.5)x || (5,2) (10,6)​ 2 = a5b 2/5b = a 6 = 2/5b (10b) 3 = 2b log2 3 = b a = .156 b = 1.585 y = .156x1.585 ||
 * ~  ||~ ** Exponential ** ||~ ** Power ** ||
 * ** Formula ** || <span style="color: #00ffff; font-family: Arial,Helvetica,sans-serif;">**// y = abx //** || //** y = axb **// ||
 * ** Example ** || (2,9) (4,20)

<span style="color: #ff8800; font-family: 'Comic Sans MS',cursive;">**Practice Problems:** Write an exponential function of the form y=ab^x that passes through the given points. 1. (1,4), (2,12) Write a power function of the form y=ax^b that passes through the given points. 1. (5,12), (7,25)

<span style="color: #000000; font-family: Arial,Helvetica,sans-serif;">**__8.8 Logistic Growth Functions__ (pg. 517 - 522)**

**P(3) = 5.808089324** ||
 * **Formula** || [[image:math_formula.jpg]] ||
 * ** Example ** || [[image:math_formula1.jpg]]

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<span style="color: #000000; font-family: Verdana,Geneva,sans-serif;">**Practice Problems:** Graph and identify asymptotes, y-intercept and point of maximum. 1. y = 5/1+e^-10x 2. y = 8/1+e^-1.02x Solve. 1. 10/1+2e^-4x= 9 2. 9/1+5e^-0.2x= 3/4


 * __WEBSITES USED:__**

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<span style="color: #ff8800; font-family: 'Comic Sans MS',cursive;">**Practice Answers: <span style="background-color: #ffffff; color: #000000; font-family: Arial,Helvetica,sans-serif;">__8.1__ ** 1. 1; x-axis 2. 3/2; x-axis 1. domain; all real numbers. range; y>0 2. domain;all real numbers. range; y>1 1. decay 2. growth 1. domain; all real numbers. range; y>0 2. domain; all real numbers. range; y>-2 1. 3e^5 2. 64/e^6 1. 5.474 2. -0.199 __**8.4**__ 1. 3 2. -0.38 1. x 1. y= e^x/6 1. 1.277 2. 1.226 __**8.6**__ 1. 1 2. approx. .3495 3. 1 __**8.7**__ 1. y=(4/3)3^x 1. y=.358x^2.181 __**8.8**__ 1. Asymptotes- x-axis, y=5 ; y-int. (0,5/2) ; pt of max- (0, 5/2) 2. Asymptotes- x-axis, y=8 ; y-int. (0,4) ; pt of max- (0,4) 1. ln18/4, about .723 2. -3.942
 * __8.2__**
 * __8.3__**
 * __8.5__**