# Tag Archives: inverse operations

## Can’t get enough of logs!

You know what fact families are? In elementary and middle school, this is increasingly common terminology to express relationships among basic number facts. A sample fact family:

$5*12=60$

$12*5=60$

$60/12=5$

$60/5=12$
Fact family thinking is intended to encourage connections between operations and their inverses. Addition and subtraction fact families work the same way as multiplication and division fact families:

$3+4=7$

$4+3=7$

$7-3=4$

$7-4=3$
I have written a lot about logarithms this past year and I keep wondering about ways to get students’ minds wrapped around the relationships that logarithms are intended to express, and to focus less on the notation.
Here is a fact family for an exponential relationship.

$3^4=81$

$\sqrt[4]{81}=3$

$\log_3{81}=4$

Some things I notice:

1. We’re not used to thinking about roots, logs and exponentials all at the same time. Maybe we should be.
2. Exponentiation is not commutative, so we only have three facts in the family, not four. That seems useful to know.
3. Off the top of my head, I can’t think of another non-commutative binary operation (besides those listed out above), so I can’t test the proposition that:
4. All non-commutative binary operations will necessarily have TWO inverse operations in their fact families.