Definition and the Basic Logarithms


  • Why Do we use Logs?

  • What are Logs?

  • The Two Basic Logs - Base 10 and Natural

    Why do we use Logs?

    In the discussion of algebraic powers it was noted that whenever a number is "raised" to a power then we write that in exponential notation and the meaning of it is that the number appearing in the base is being multiplied by itself the number of times that is indicated by the exponent. The notation used was such that if we write 53, what we actually mean is "5 multiplied by itself 3 times".

    Logarithms are mathematical inventions in order to answer a slightly different question (notice the word "invention"; logarithms make certain operations easier to handle and that is all they do, so you should think of them as a definition). In order to motivate why logarithms are introduced in the first place, let us invent a scenario. Suppose someone asked you the following question:

    Now, this is easy to answer by thinking about powers because the above example is simple powers and simple number, and once can reason it out relatively quickly. However, things can get more complicated. Suppose nowt that you were asked "what number do I have to raise 10 to in order to get 735?" - and you will find yourself asking such questions in certain contexts in astronomy! All of a sudden the answer is not very obvious. What is so different about this question?

    There is actually nothing different about this question. You still can try doing the same process, but now the nubmer isn't that pretty and it's not exactly obvious how many times you should multiply 10 by itself to get 735. If you multiply it by itself 2 times you get 100, but 3 times gives 1,000, and you have already exceeded 735! How do we "get out" this power that we need?

    Logarithms are - at the most basic level - invented to answer the more general question of how does one extract the base or exponent of an algebraic power when one of these is an unknown.

    What are Logs? The Definition

    Continuing on the above reasoning, let us take our simple example again: what number raised to the power of 3 gives 1,000? If we invent an unknown variable - call it x- and try to write out our question in terms of the notation of algebraic powers we have the following situation:

    10x = 1,000

    The question is: what is x in the above formula? how do we solve for x? We invent an operation called the logarithm - abberviated as Log - and we apply this operation to both sides of the above relation.

    Log10x = Log1,000

    We then define this new logarithm function to be such that when it applies to a number that is "10 raised to some power" then it literally just gives us the power and the the base 10 "disappears". To reiterate this, what we mean is the following: Log10x = x .

    So, what is the value of Log1000 ? How do we know what the right hand side of the equation is? How does this help us at all. Well, this will probably seem magic to you at this point, but one option is to find the [Log] button on your calculator and take the "Log(1000)", and ... you get 3!! So, x = 3, and it is what we expected!

    How does this help us with anything? It seems like we went in a big loop, and we knew the answer to begin with anyway! But ... now consider the slightly more complicated question that we had above: "what number do I raise 10 to in order to get 735?". Let us apply the logarithmic process to this situation:

    If you take the Log of 735 on your calculator you get, 2.866...! So, 10 raised to the power of 2.866... gives you 735, and the question is answered. Recall that algebraic powers need NOT be intergers, and here we have a clear example of a non-integer power.

    The Two Basic Logs - Base 10 and Natural

    There are two basic types of logarithms that are important to know for Astronomy (and basically for most Sciences as well). In the previous section, where logarithms were defined, you already saw the definition of one kind of logarithms; that was the so called "log base 10".

    The logarithmic operation that we have introduced serves the main purpose of extracting the exponents in an algebraic power. This is true of the operation of "taking the logarithm".

    Logorithm Base-10

    The logarithm of base 10 is most often useful when powers of 10 are involved, but not necessarily. It can be used in many other situations. For instance, suppose you were asked the following question: 3 raised to what power gives 16.8? Again, applying our definition of logarithm of base 10 - as defined in the previous section - we can answer this question ... but, in order to do this we need to define some rules of operation for logarithms (this is outlined in the next page on logarithms).

    You can think if Logarithm Base-10 as the logarithmic operation that when carried out on 10 raised to some power ends up giving us the power. The log of base 10 is written as: Log10. Thus, Log10(10x) = x. This is the basic definition of base 10 logs.

    Natural Logarithm (or base e)

    There is another logarithm that is also useful (and in fact more common in natural processes). Many natural phenomeno are seen to exhibit changes that are either exponentially decaying (radioactive decay for instance) or exponentially increasing (population growth for example). These exponentially changing functions are written as ea, where a represents the rate of the exponential change (see the section on graphing to refresh your memory of exponential functions).

    In such cases where exponential changes are involved we usually use another kind of logarithm called natural logarithm. The natural log can be thought of as Logarithm Base-e. What this means is that it is a logarithmic operation that when carried out on e raised to some power gives us the power itself. This logarithm is labeled with Ln (for "natural log") and its definition is: Ln(ex) = x.