Exponential+&+Logarithmic+Functions

__**What is the general form of an exponential function?**__
Exponential functions have the general form:
 * //y// = //f// (//x//) = //a// x ****,** where //a// **> 0** **, //a//≠1 **, and ** //x// ****is any real number**.

The reason why //a// > 0 is positive is because by keeping //a// values positive, it allows the function to have a domain of all real numbers. In addition, //a// is called the base of the exponential function for all equations.

The main rules used in manipulating exponential functions are:
 * bx+y = (bx)(by)
 * bxy = (bx)y
 * b0= 1
 * b-x = 1/(bx)

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__**What is the natural exponential function?**__
The graph of The function// **y** //** = // e // //x// ** slopes upward, and increases exponentially as //x// increases. The graph also always lies above the //x//-axis but can get close to it for negative //x// values (As seen in the illustration). Thus, the //x//-axis is considered to be a horizontal asymptote as it acts as a cut-off or hole for the y values. In simpler terms, if the x-axis is the horizontal asymptote the no y values of the function will ever be 0
 * // y // = // e // //x// **

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__** What is e^x? **__
In mathematics, the exponential function is the function e^x, when e is the number (approximately 2.718281828) such that the function e^x is its own derivative.

The base can be any positive number (except one). However, two choices are most usual: 10 and e= 2.718281828…

Logs to the base are often called common logs, whereas logs to the base e are often called natural logs.

__** Properties of Logarithms **__
Property 3: Multiplication inside the log can be turned into addition outside the log, and vice versa. Property 4: Division inside the log can be turned into subtraction outside the log, and vice versa.

Property 5: An exponent on everything inside the log can be moved out fron as a multiplier, and vice versa.



Examples:


 * __Expand__ log3(x2y7) **


 * log3(x2y7) = log3x2 + log3y7 **


 * = 2log3x + 7log3y **

= 2log3x + 7log3y

__** Expand **__** log4(4x/y9) **


 * log4(4x/y9) = log44 + log4x – log4y9 **


 * = ** log44 + log4x – 9log4y

media type="youtube" key="pP3NunYYhzk" width="425" height="350" media type="youtube" key="MmJ5z_ZP1kU" width="425" height="350" http://www.mesacc.edu/~scotz47781/mat120/notes/expand_logs/expanding_logs_intro.pdf http://www.youtube.com/watch?v=pP3NunYYhzk http://www.purplemath.com/modules/logrules.htm = =

=__How are f(x) ln(x) and g(x)__= e^x related?== These functions of the inverse of each other and can be used when solving problems to find a variable. This also means that the graph are symmetrical about y=x.

This video below is an excellent example of two ways to solve one problem. One way is using the properties of natural logs and the other is using e to get rid of the natural log. media type="custom" key="20759452" width="128" height="128" align="left"

What are the domain and range of f(x)= ln(x) and g(x)= e^x?
The domain of ln(x) is (0, infinity) this means that all positive real numbers are in the domain of the function ln(x). The range of this function is (-infinity, infinity). In the graph ln(x) there is an asymptote at x=0. Which means that the y values can go from negative infinity to positive infinity but the x values are all positive real numbers. This means that the domain and range are flipped for the function e^x because these two functions are inverses. Therefore the domain of e^x (-infinity, infinity ) and the range is all positive real numbers like the domain of ln(x).
 * ~ x ||~ y=ln(x) ||
 * 0 || ln(0)=undefined ||
 * 2.72 || ln(2.72)=1 ||
 * 7.39 || ln(7.39)=2 ||
 * 20.09 || ln(20.09)=3 ||


 * ~ x ||~ y=e^x ||
 * 0 || e^0=1 ||
 * 1 || e^1=2.72 ||
 * 2 || e^2=7.39 ||
 * 3 || e^3=20.09 ||

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=**7. How do you solve equations involving logarithms and exponential functions?**= The first thing you need to know is how to solve exponential functions.
 * Some problems are extremely simple. For example, solve for x: 5^x=5^3. It's as simple as x=3
 * This same concept can be used for problems that are a little more complicated: 10^(1-x)=10^4. So (1-x) = 4, so -x = 3, so x=-3.
 * Sometimes the bases won't be the same, so you have to convert them to be equal to each other. The basic way is thinking of it like this: 3^x=9 -> 3^x=3^2 -> x=2
 * So now when you have a problem like this: 3^(5x-2)=27, you can solve it like this: 3^(5x-2)=3^3 -> 5x-2=3 -> 5x=5 -> x=1
 * Problems dealing with fractions are just the same as long as you remember that negative exponents get you those fractions: 4^x=1/64 -> 4^x=1/(4^3) -> 4^x=4^-3 -> x=-3
 * The same is for roots as well, since the squareroot of x is the same as x^1/2. So, 8^x=sqrt(8) is the same as 8^x=8^1/2. x=1/2

These are pretty basic problems, but you can use the concepts to solve more complicated ones. In order to solve exponential equations where the bases cannot be converted to be equal to each other, you must know how to solve logarithms, the inverse of exponents. Solving logs is just like solving problems with exponentials. Here are the basics for solving logs you need to remember:
 * //log// b(//x//) = //y// means the same thing as b^ //y// = //x,// this being said, logarithms are just exponents written differently.
 * //log// b(b) = 1, for any base b, because b^1 = b.
 * //log// b(1) = 0, for any base b, because b^0 = 1.
 * //log// b(//a//) is undefined if //a// is negative.
 * //log// b(0) is undefined for any base b.
 * //log// b(b //n// ) = //n//, for any base b.
 * Use the properties of logs to simplify logs as well.

Here’s some examples on how to solve a lot of the different types of logarithm problems:

 * Solve log2(8)=y. This equation means the same thing as 2^y=8. Since 2^3=8, than y must be equal to 3.
 * Simplify log64(4). So we're trying to find what exponent would make 64 = 4. We know that 4^3 = 64, so we then know that 64^1/3 = 4. Therefore, log64(4) = 1/3
 * Simplify log4(–16). This problem __** cannot **__ be simplified because there is 4 to the power of any number cannot equal a negative number.
 * Simplify logb(b^3). This can translate into b^y=b^3, so logb(b^3) = 3. The base b and the other b don't technically cancel each other out, but it's a useful way of thinking about it.

To solve problems with exponentials where the bases aren't equal and can't be converted to each other, we use logs.

 * 2^x=30 is one of those problems. You can find x just by changing 2^x=30 into log2(30)=x. To get an answer from your calculator, you have to apply the change of base formula into your calculator. This would make x= ln(30)/ln(2). Of course, the answer log2(30) is much simpler.

=**8. How do you graph f(x)=ln(x) and g(x)=e^x?**= f(x) = ln(x) and g(x) = e^x are unique graphs. Further up on the page, you see the two graphs compared in order to show their relationship as inverses: Remember that when graphing logs, x is only defined when x>0.

The transformations that can occur in these graphs are similar to what you've seen on other graphs. The video below explains how to graph f(x) = 3ln(x-2) + 4 without using a calculator. media type="youtube" key="Do3nZymgRHw" height="315" width="560"