Question

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?

Answer 1

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`