How does logs work

There is a change of base formula for converting between different bases. To find the log base a, where a is presumably some number other than 10 or e , otherwise you would just use the calculator,. There is no need that either base 10 or base e be used, but since those are the two you have on your calculator, those are probably the two that you're going to use the most.

I prefer the natural log ln is only 2 letters while log is 3, plus there's the extra benefit that I know about from calculus. The base that you use doesn't matter, only that you use the same base for both the numerator and the denominator. Therefore, the value of this logarithm is,. Before moving on to the next part notice that the base on these is a very important piece of notation. Changing the base will change the answer and so we always need to keep track of the base.

First, notice that the only way that we can raise an integer to an integer power and get a fraction as an answer is for the exponent to be negative. So, we know that the exponent has to be negative. In this case if we cube 5 we will get Now, just like the previous part, the only way that this is going to work out is if the exponent is negative.

Hopefully, you now have an idea on how to evaluate logarithms and are starting to get a grasp on the notation. There are a few more evaluations that we want to do however, we need to introduce some special logarithms that occur on a very regular basis. They are the common logarithm and the natural logarithm. Here are the definitions and notations that we will be using for these two logarithms.

To do the first four evaluations we just need to remember what the notation for these are and what base is implied by the notation. Notice that with this one we are really just acknowledging a change of notation from fractional exponent into radical form. This example has two points. First, it will familiarize us with the graphs of the two logarithms that we are most likely to see in other classes. Also, it will give us some practice using our calculator to evaluate these logarithms because the reality is that is how we will need to do most of these evaluations.

Properties 3 and 4 leads to a nice relationship between the logarithm and exponential function. Recall from the section on inverse functions that this means that the exponential and logarithm functions are inverses of each other. This is a nice fact to remember on occasion. We will be looking at this property in detail in a couple of sections. We will just need to be careful with these properties and make sure to use them correctly.

Relate the power rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of powers. We have already seen that the logarithm of a product is the sum of the logarithms of the factors:. Then we have:. The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities.

Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients. A logarithm written in one base can be converted to an equal quantity written in a different base. Fortunately, there is a change of base formula that can help. The change-of-base formula can be applied to it:. Exponents and Logarithms work well together because they "undo" each other so long as the base "a" is the same :.

They are " Inverse Functions ". It is too bad they are written so differently So it may help to think of a x as "up" and log a x as "down":. The Logarithmic Function is "undone" by the Exponential Function.

Using that property and the Laws of Exponents we get these useful properties:. History: Logarithms were very useful before calculators were invented That is as far as we can simplify it