FSTChapter+6

toc media type="custom" key="640819" =Chapter 6: Root, Power, and Logarithm Functions=

6-1: //n//th Root Functions


media type="custom" key="516827" media type="custom" key="584247" nth root-when //n// is an an interger __>__ 2, x is the nth root of k IFF x^n=k power of a power property-for any no neg. base x &amp; any nonzero real exponents m & n (x^m)^n=x^mn example: (x^2)^3=x^6 this chapter is mostly things we have learned before but some other things to know are: up to 2 answers with even exponents and only 1 answer with odd the exponent of 1/n is the inverse exponent of n when these 2 are graphed the are reflection images over the line y=x because of their inverse relationship, every formula with nth powers can lead to the calculation of nth roots
 * __vocabulary to know:__**

Alyssa W. and Evan M.

6-2: Rational Power Functions


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Any function //f// with equation of the form f(x) = x^(m/n), where m and n are nonzero integers, is a rational power function.

__Rational Exponent Theorem:__ For all positive integers //m// and //n,// and all real number x&gt;=0, x^(m/n) = (x^1/n)^m, the //m//th power of the positive //n//th root of x, and x^(m/n) = (x^m)^(1/n), the positive //n//th root of the //m//th power of x

This theorem shows us two different ways that we can break down a number raised to a rational exponent (in a fraction).

Here are some previously learned properties involving exponents that are reviewed in this lesson. For any nonnegative bases x and y and nonzero exponents, or any nonzero bases and integer exponents: __Product of Powers Property:__ x^m * x^n = x^(m+n) (* means multiplied by) __Power of a Product Property:__ (xy)^n = x^n * y^n __Quotient of Powers Property:__ x^m / x^n = x^(m-n) (x doesn't = 0) __Power of a Quotient Property:__ (x/y)^n = X^n / y^n (y doesn't = 0) __Zero Exponent Theorem:__ x^0 = 1 (If x is any nonzero real number) __Negative Exponent Theorem:__ x^ -n = 1 / (x^n) (For all x&gt;0 and n a real number, or for all x doesn't = 0 and n an integer)

__Graphs of rational power functions__: Domain: x>=0 Range: x>=0 examples can be found in your book on page 379

__Example:__ Evaluate 27^( -5/3). Give an exact answer. 27^( -5/3) = (27 ^ (1/3)) ^ -5 = 1 / ((cube root of 27) ^ 5) = 1 / (3 ^5) = 1 / 243

-Austin B.

This section talks about rational power functions and theorems about exponents. It states the Rational Exponent Theorem which is (x^m/n=(x^1/n)^m=(nth root of x)^m and (x^m/n)=(x^m)^1/n=nth root of x^m. Some other properties that were given were Product of Powers Property (x^m //x// x^n = x^m+n), Power of Product Property ((xy)^n = x^ny^n), Quotient of Powers Property (x^m/x^n = x^m-n when x does not = 0), and Power of a Quotient ((x/y)^n = x^n/y^n when y does not equal 0). It also stated that when x does not equal 0, x^0=1 and when x>0, x^-n=1/x^n. The lesson ended by sayin that when graphing a rational power function the domain is x greater than or equal to 0, and so is the range.

-Travis D.

6-3: Logarithm Functions


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Logarithm: //b// &gt; 0 and //b// does not epual 1, then //y// is the loaruth of //x// to the base of //b// y=logbx IFF b^y=x REMEMBER!!! The //b// is the base in both equations and //y// is the exponent. A logarithm with a base of 10 is known as a common logarithm. 7 Properties of Logs: -Katie S.
 * The domain is the set of all positive real numbers
 * The range is the set of all real numbers
 * The point (1,0) will be on all graphs; log//b//1=0
 * The function is strictly increasing; it will never go down.
 * As //x// increases, //y// has no bounds
 * When //x// is between 0 and 1, the eqponent will be negetive
 * The //y// axis is an asymptote

 Definition of logarithm: Let b>0 and b does not equal 1. Then y is the logarithm of x to the base b, written y = logbx, if and only if b^y=x. Common logarithms: Logarithms with base of 10. Properties of Logarithms: 1. Domain is the set of positive real numbers 2. Range is the set of all real numbers 3. (1,0) will be on the graph 4. The function is strickly increasing 5. As x increases, y has no bound. 6. When x is between 0 + 1, the exponent will be negative. 7. The y-axis is an asymptote The function which maps x onto logbx is called the logarithm function with base b. Common logarithms are often written without indicating the base 10. -Jesse B.

6-4: //e// and Natural Logarithms


media type="custom" key="557473" When compounding continuously you reach the number "e". When compounding continuously you can use the function A(t)=Pe^rt. This is the continuous change model. Also the natural logarithm of x is the logarithm to the base e, written In x. That is, for all x, In x=logex.

Troy pretty much covered the rest Thank u Lantz

In section 6-4 we are learning and interpreting the functions of "e" being continuously compounded. The e button on your calculator can be found by pressing 2nd divided sign. e is the value of 1 compounded continuously and is approximately 2.718281824590... This value is irrational and goes on forever.

Remember when compounding continuously, you can use the function A(t)= Pe^rt, which is the Continuous Change Model where "P" is the principle that can grow or decay continuously, "r" is the annual rate, "t" is the number of years, and A(t) is the amount after t years.

The exponential function with base e is f(x)=e^x this can also be written in Logarithm form as a natural logarithm x=log(e)x using our function f(n)((x))=(1+1/n)^nx we noticed that as "n" increases our graph curves closer to f(x)=e^x

The last studied property was the inverse of this function written f^-1(x)=log(e)x the inverse of natural logarithm functions is the reflection over the line y=x on your calculator the natural logarithm is "ln", as "log" only covers common logarithms. Thank you for your time, if you have any questions about my explanation please ask, I hope that I have cleared up any confusion about this lesson. Sincerely, Troy L

6-5: Properties of Logarithms


media type="custom" key="552415" In lesson 6-5 we learned about properties of logarithms. The first theorem is //Logarithm of 1//- this is when if you have a log base b 1 it will always equal 0: Logb 1 The next theorem is //Logarithm of a Product//- in this one if you have Logb(xy) it = Logb x + Logb y it is used when you have 2 logs with the same base and you want to combine them or if you have 2 numbers multiplied together in a log you can split them into 2 logs. Next is the Logarithm of a Quotient- this theorem is when you have a division problem in a log, and you want to split it up: Logb(x/y)= Logb x-Logb y. Finally is the theorem of Logarithm of a power. This is when x in the log of x has an exponent such as: Logb x^p and for this you can move the exponent in front of the Log: pLogb x. Robert W.

The Theorem for Logarithin of 1 means that any log with the base of 1 is going to equal 0. For any base b and for any positive real numbers x and y, the log base of b will be log b x + log b y. Katie K.

6-6: Solving Exponential Equations
Online lesson for Section 6-6

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This lesson is about solving exponential equations or equations where the exponent is the variable. For example: 3^x = 15. A technique to solve this is to use a graph and find the point using the tracing feature on your calculator. Another technique is using log. Lets go back to the equation of 3^x = 15.

-To solve, take the log of each side: log(3^x) = log 15 -use the log of power theorem to move x in front of the log: x log 3 = log 15 -divide each side by 3 x = (log 15 / log 3) Just enter it into your calculator and you get about 2.4650

This can also be a useful technique in solving natural logs. Lets use the equation 110 = 94e^(0.019t.)


 * MAKE SURE TO USE PARENTHESIS AROUND THE EXPONENT!!!!!!!

-divide each side by 94 to get e by itself (110/94) = e^(0.019t) -take the natural log of each side to get rid of e, this leaves you with 0.019t ln(110/94) = 0.019t -divide by 0.019 to get t by itself (ln(110/94))/0.019 = t -t is about 8.273

This is a useful technique that can be used on natural logs and exponential equations

--ANDREW C. & AUSTIN A.--

6-7: Linearizing Data to Find Models
Online lesson for Section 6-7 In lesson 6-7 we learned about linearizing data to find models. Sum this whole chapter up you can take the volume, a cube, joules, anything and break it down into data then get a graph. But, it doesn't stop there if it is a curved graph the linearizing meaning that they made the graph into a line and set it equal to something. These models help us get our data quicker cause all you have to do is go over the row or follow the line and get your answers. That simple making anything into a line graph. ;) Yea booooyy! Zack L. Peace Out! media type="custom" key="573195"

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