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.