Why do you need to invent a whole new set of mathematical notation, logs, if you could just express the same answer in quicker, less "complex" exponential form?

1 Answer
Aug 6, 2018

It is notationally much more convenient in circumstances where logarithms naturally arise. Using exponential form in such circumstances can result in convoluted descriptions.

Explanation:

Suppose you were trying to describe what the integral of #1/x# is.

Using the natural logarithm, you can write:

#int \ 1/x \ dx = ln abs(x) + "constant"#

In words: "The indefinite integral of one over #x# is the natural logarithm of the absolute value of #x# plus the constant of integration".

If you expressed this in exponential form, you would write something like:

#e^(int 1/x dx) = Cx#

Does this describe what the integral of #1/x# is?

Yes, but only in a convoluted way:

In words: "The indefinite integral of one over #x# is an expression in #x# such that when you take the natural exponent gives an constant multiple of #x#".

More simply, suppose you were asked to find the solution of:

#e^x = 2#

Without logarithms, you could answer: It is an irrational number, approximately equal to #0.693147#, whose exponent is #2#.

With logarithms you can say #x = ln 2#

Often, re-expressing equations that use logarithms in terms of descriptions that do not effectively becomes a convoluted way of speaking about the inverse of exponentiation - i.e. logarithm.

For example, try reformulating the following without logarithms:

#(ln x) (ln y) = c#