2.4 Exponential Function Manipulation

Rewriting Exponential Expressions

Exponential functions can be rewritten using properties of exponents, like $a^{x+y}=a^x\cdot a^y$ and $a^{xy}=(a^x)^y$. This lets you convert between forms such as $f(x)=b^{kx}$ and $f(x)=(b^k)^x$, revealing a different growth/decay factor while representing the same function. Rewriting helps identify percent change per unit interval.

Equivalent Bases and Growth Rates

Any exponential function $a\cdot b^x$ can be rewritten with a different base by expressing $b$ as a power: $b=c^k$, so $a\cdot b^x = a\cdot c^{kx}$. This is useful for comparing growth rates, converting between continuous ($e^{kt}$) and discrete growth models, or matching a desired time unit (e.g., converting an annual rate to a monthly rate).

Why Manipulation Matters

Different algebraic forms of the same exponential function highlight different features: standard form shows initial value and growth factor per unit, while rewritten forms can reveal growth factor over a different time period (e.g., per week instead of per day). Manipulation doesn't change the function's output — only how the rate of change is expressed.

Key terms

Rewrite $a^{xy}$

$a^{xy} = (a^x)^y = (a^y)^x$ — used to change the base of an exponential expression.

Why rewrite exponential functions?

To reveal growth/decay rates over different time intervals or to convert between bases (e.g., base $b$ to base $e$).

Convert base $b$ to base $e$

$b^x = e^{(\ln b)x}$, since $b = e^{\ln b}$.

Per-unit growth factor

In $a\cdot b^{x/n}$, the growth factor per unit $x$ is $b^{1/n}$, found by rewriting as $a\cdot(b^{1/n})^x$.

Does rewriting change the function's values?

No — equivalent algebraic forms represent the same function, just expressed with a different base or exponent structure.

Try an AP-style question

Which is equivalent to 2^(x + 3)?

1. 2ˣ + 8
2. 6 · 2ˣ
3. 8 · 2ˣ
4. 2 · 3ˣ