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