Wednesday’s Why Episode 6
In this week’s episode of
Wednesday’s Why we will tackle why the following is “law” of exponents is true
in a way that hopefully will promote mathematical fluency and confidence. It is my hope that through these Wednesday’s
Why episodes that you are empowered to seek deeper understanding by seeing that
math is a written language and that by substituting equivalent expressions we
can manipulate things to find truths.
Now of course the base of
2 is arbitrary, but we will use a base of two to explore this.
The first thing to be
aware of is that exponents and logarithms deal with the same issue of repeated
multiplication. There connection between
the properties of each are tightly related.
What we will see here is that the property of logarithms above and this
property of exponents below are both at play here. But it is not so easy to see, so let’s do a
Just to be sure of how
exponents and logarithms are written with the same meaning, consider the
Let us begin with
We can rewrite this as a logarithm,
We can rewrite this as a
logarithm, statement 2.1:
If we take the product of M and N, we would get 2A
exponents are repeated multiplication,
·2B = 2A + B
This gives us statement
Let us rewrite statement
3 as a logarithmic equation.
In statements 1.1 and 2.1
we see what A and B equal.
So let’s substitute those now.
It took a little algebraic-juggling
to get it done, but hopefully you can now see that this is not a law or a rule,
but a property of repeated multiplication, just like all of the properties of
exponents are consequences of repeated multiplication.
Let me know what worked
for you here and what did not. Leave me
Thank you again.