Twin Powers

Learning Focus

Connect related exponential situations: equivalent forms with different bases, and growth vs. decay "twins."

Big idea

• Change of base: $4^x=(2^2)^x=2^{2x}$. Doubling every unit is the same as quadrupling every two units.
• Growth/decay twins: $\left(\tfrac12\right)^x=2^{-x}$. The graph of $2^{-x}$ is the reflection of $2^x$ across the $y$-axis.
• A growth factor $b$ and decay factor $\tfrac1b$ describe the same process run forwards or backwards.

Same situation, different time units $\Rightarrow$ different bases that give identical outputs.

Worked example

A culture starts with $3$ bacteria and doubles every hour: $N(t)=3\cdot2^t$.

• After 2 hours: $3\cdot4=12$
• It quadruples every 2 hours, so equivalently $N=3\cdot4^{t/2}$

Check at $t=2$: $3\cdot4^1=12$ ✓.

Decay twin: $M(t)=3\left(\tfrac12\right)^t=3\cdot2^{-t}$.

Takeaways

• Use $(b^m)^n=b^{mn}$ to rewrite bases.
• $\left(\tfrac1b\right)^x=b^{-x}$ reflects growth into decay.
• Match the base to the time unit you are using.