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What is the change of base formula?
In math, that may either refer to changing the base of the number system (for example, change from decimal (base 10) to binary (base 2)); or it may refer to changing logarithms, from one base to another - for example, common (base-10) logarithms to natural (base-e) logarithms.
What is the logarithm of 2?
log 2 = 0.30102999566398119521373889472449 for base 10 logarithms
Why natural log has base e?
Because when the system of logarithms with the base 'e' was defined and tabulated, it was entitled with the identifying label of "Natural Logarithms". ---------------------------------- My improvement: The natural log base is e (a numerical constant of about 2.718). It is chosen as a log base since there is a mathematical series (a "string" of mathematical numerical terms to be summed) for calculating a logarithm (ie. exponent of the base) of a number, which has a base of e. Series for calculating logarithms with bases other than e have basically not been developed.
How do you compare logarithms with different bases?
You can convert to the same base, by the identity: logab = log b / log a (where the latter two logs are in any base, but both in the same base).
What does Ln?
ln stands for the function that associates a value with it natural logarithm or, in other words, its logarithm to the base e. You are probably familiar with common or base 10 logarithms and know that, for instance, log10100 = 2 because 100 = 102. ln works in the same way. loge e2 = 2. The value of e is about 2.71828. Therefore, loge 2.71828 ~=1. This function has characteristics that parallel those of base 10 logarithms. You might wish to see the wikipedia page about the natural logarith.
