Why Logarithms & Radicals Are Glorified Division

Note: You can also see this cartoon image somewhere in the Still Image/Pocket Joke Section.

This math trick will explain why logarithms & radicals are both glorified division. Out of all 4 arithmetic operations, division is the one that has 2 types of glorification!

For example, suppose that you want to know what 225 (2 to the 25th power) is. Then you can either factor the exponent like this:

225 = 212 * 213 = 4,096 * 8,192 = 33,554,432

212 = 4,096; 213 = 8,192

12 + 13 = 25 since multiplying with the same base adds the exponents up; in other words, ab * ac = ab + c

So, log2(33,554,432) = 25

(Dividing 33,554,432 by 2 exactly 25 times in a row will get you to 1)

(logab = c simply means that you have to divide the variable b by the variable a exactly c times to get to 1.)

2 to the 25th power is a really large number, isn't it? Well, if the value of b was equal to 8, then...

8 ÷ 2 = 4; 4 ÷ 2 = 2; 2 ÷ 2 = 1

Which is why log2(8) = 3

(It took 3 divisions by 2 in a row to get from 8 to 1!)

That shows that logarithms are glorified division!

...Or you can factor the exponent like this:

225 = (25)5 = 325 = 33,554,432

25 = 32; 5 * 5 = 25

Parentheses cause exponents to multiply! To describe it, (ab)c = abc (or ab * c) Only 1 copy of the base is needed.

However, this proves that 5√(225) = 25

And this 2nd example shows that radicals are glorified division, too! In fact, radicals are fractions in exponential form! Remember this radical rule:

Fractions are mainly about division, especially with integers! Radical signs can be used to divide exponents literally! Finally, as I conclude this Web page, math problems with logarithms or radicals are very likely to have irrational numbers as correct answers!

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© Derek Cumberbatch