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Logarithmic Functions

By Edited Nov 13, 2013 0 0

There are several different kinds of logarithmic functions. As you should know, all logarithms have a base. So every different base is essentially a new logarithmic function. The three most popular bases are 10, 2, and 2.718. The last number listed is called Euler's number and is what defines the natural logarithm.

The number e is the most convenient choice of base for a logarithmic function. There are several reasons for this, but it is mainly because it is easy to take its derivative. Essentially taking a derivative with this base normalized the constant to one. There is special notation for a logarithmic function with the base e;

loge (x) = ln (x)

This basically says that if we have a logarithmic function with the base, then we replace it with the function 'ln'.

There are few defining properties of the natural logarithmic function. You should know these in order to truly understand how the function works.

lnx = y <-> e^y = x

This basically says that the ln function is the inverse of the natural exponential function.

ln(e^x) = x

e^(lnx) = x

In both cases above we have two inverse functions acting on x. In both cases the function affect will cancel out and we will simply be left with x.

It is also very important to note that if we try and take the natural logarithm of the number e we will simply be left with one. This is the same affect, where the inverse functions cancel each other out. You can think of this is the case where x=1.

Now let's say we are asked to solve some logarithmic functions. How would you go about doing this? It is the inverse of an exponential function so it is not necessarily easy to do. You must do some manipulation and rearranging before you can solve the equation.

Say we are given lnx = 2 and we are asked to solve the value of x.

Using the first property listed above we can rearrange this equation so that we can solve for X. We can say that if lnx =2 then e^2 = x. Therefore we can say that x = e^2. We simply applied the inverse function formula to solve the logarithm.

Solving logarithmic functions is not always easy. Having several different bases can get rather complicated. It will take some practice to get them down. But don't give up, go find more examples of logarithmic functions.


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