Logs and the Power They Wield

Logarithmic functions are in their most basic from when they look something like this:

This particular log has a base of b and an argument of a. It is pronounced "log-base b of a equals x" and it is equivalent to saying:

Though, remember that the two equations do NOT equal each other, rather they are interchangeable, and note that a can never be less then or equal to 0, otherwise it is impossible to solve without imaginary numbers.

To go about solving a logarithm such as this one:

We first have to set it equal to something so that it turns into an equation. In this case, we will use x as a variable that it equals. So:

From here, we can apply the previous method and solve for x.

So:

If you ever come across something like this:

 or this:

Don't panic, they are just a shorthand way of writing different bases.

The first one means there is a base of 10, or, in other words "log-base 10 of a".

The second one means there is a base of e, or "Natural log of a".                

The last major thing to know about logarithms is their graph. You may have noticed that a logarithmic function is the inverse of a simple exponential function, and, because of this, the graph of a logarithmic function is an exponential function reflected on y=x. This also means that the range of the exponential function is the domain of the logarithmic function that is its inverse, and vis versa.

The major components of the parent function:

is as follows:

    Domain: ( 0, ∞)

    * Range: (-∞,∞)

    x-intercept: (1,0)

    y-intercept: N/A

    vertical asymptote: 0

    * horizontal asymptote: none

* These will be true for any logarithmic function

You can use these components to find the graph of any log function by finding the shifts in the horizontal and vertical and by plugging in 0 for x and y, just as you would any other graph.

Properties of Logarithms

There are three properties of logarithms:

Also, you do not need to only use  ; you can use  , ,   , and more

Here is one way to prove 

First, you can set up the equation:

Then, subtract logb from both sides:

Next, using the definition of a logarithmic function you can change it into this form:

Subtraction in the exponents means you can change it to this:

The denominator can then simplify:

You can multiply both sides by b:

You can then take the log of both sides:

Since , 

You can use these properties to expand or simplify/compress logarithmic expressions:

Expanding:

Simplifying:

There is also the change of base formula:

To prove this formula:

First, you can set up the equation: 

and change it to:

Next, take the log of both sides; it does not have to be base 10:

Using the properties of logarithms:

 

Divide:

 

For example, if you had , you could use the change of base formula to change it to or even . 

One common mistake that people make its they think log(a+b) = loga + logb. This is NOT true! 

Rational Equations

A rational equation is an equation that can be written as

 Where N(x) and D(x) are polynomials.

 

 The Domain of a rational function includes all real numbers except for any values where D(x)=0.  this is because if D(x)=0, The function would be undefined.  However, the zeros of D(x) are still important to rational functions.  At the zeros of D(x), there is an imaginary line known as a vertical asymptote.  As the Graph approaches the asymptote, the Y values will go to infinity and negative infinity but never cross the line where D(x)=0.  

Graphs can also have horizontal asymptotes.   

If the degree of N(x) is Greater than the degree of D(x), then there is no horizontal asymptote.

If the degree of N(x) is less than the degree of D(x), then there is a horizontal asymptote at Y=0.

If the degree of N(x) is equal to the degree of D(x),then the horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and denominator.

Other points of interest include:

X-intercepts: when f(x)=0 or more simply, when N(x)=0.

Y-intercept: f(0)

Example:

Based on the equation, we can determine that there will be a vertical asymptote at x=3 and a horizontal asymptote at y=2.
By solving for f(0), we can determine that the y-intercept will be at (0, 0)
By solving for f(x)=0, we can determine that there is an x-intercept at (0, 0)

In order to describe what the graph is doing at the vertical asymptote, we must use limit notation.

Saying that the graph decreases as it moves toward the Asymptote from the left, we say

The Superscript - on the 3 indicates the left side of the graph.
Thus in order to describe the Right side of the graph we use a superscript + on the 3.

Fundamental Theory of Algebra

The fundamental theory of algebra states that any polynomial of degree n has n roots. For example, the function  has seven roots, because the highest exponent is seven. Those roots are . Five of these roots are real, but two are unreal. This can be seen in the graph:

Again, this function has five, real, visible roots and two unreal ones that cannot be easily seen on the graph. The roots we cannot see as x-intercepts can be assumed to be imaginary. This can be demonstrated in the translation of the function .

Four real roots, zero imaginary roots. 

Two reals roots, two imaginary roots.

Zero real roots, four imaginary roots.

As you probably have noticed, the roots in this function come in pairs. Real roots only come in pairs in a function that is symmetrical about the y-axis, but imaginary roots always come in pairs, no matter the shape. This is because imaginary roots come in conjugate pairs. For example, if F(2+3i)=0 then, F(2-3i)=0.