Topic 2.4 Notes – Exponential Function Manipulation

What Equivalent Exponential Forms Are

Two forms are equivalent when they match for every real input. Example:

$$3^{x+2}=9\cdot 3^x$$

Those are the same function, just written to show different features.

For AP Precalculus exponential functions, the base must satisfy

$$b>0 \quad \text{and} \quad b\ne 1$$

That keeps $b^x$ real for all real $x$, and it avoids the constant function $1$.

You’ll see the same function written in forms like these:

• $b^{x+k}$ shows a horizontal translation
• $ab^x$ shows a vertical multiplier
• $b^{cx}$ shows a horizontal scaling (and maybe reflection)
• $(b^c)^x$ shows a changed base

Quick graph reminder. Changes inside the exponent affect inputs horizontally. A coefficient outside affects outputs vertically. In the graph, $y=3^{x+2}$ and $y=9\cdot 3^x$ land on the same curve, which is exactly what equivalent forms mean.

Equivalent exponential forms

The Exponent Rules That Drive the Rewrites

Product property

$$b^m\cdot b^n=b^{m+n} \qquad \text{and} \qquad b^{m+n}=b^m b^n$$

This only works with the same base.

Power property

$$(b^n)^m=b^{nm}$$

Reverse use matters a lot here:

$$b^{cx}=(b^c)^x$$

Exponents multiply only in a power of a power situation.

Negative exponent property

$$b^{-n}=\frac{1}{b^n}$$

A negative exponent means reciprocal, not a negative number.

Unit-fraction and rational exponents

$$b^{1/k}=\sqrt[k]{b}$$

$$b^{m/k}=\left(\sqrt[k]{b}\right)^m=\sqrt[k]{b^m}$$

$$b^{-m/k}=\frac{1}{b^{m/k}}$$

Compact examples

Rewriting Exponential Functions to Show Transformations

A horizontal shift in an exponential can be rewritten as a vertical multiplier:

$$b^{x+k}=b^k b^x$$

So $k>0$ shifts left, and that same change becomes multiplying outputs by $b^k$.

• $5^{x-3}=5^x\cdot 5^{-3}=\frac{1}{125}5^x$
• $4^{x-2}=\frac{1}{16}4^x$

If there’s already a coefficient,

$$ab^{x+k}=(ab^k)b^x$$

A horizontal scaling can be rewritten as a new base:

$$b^{cx}=(b^c)^x,\quad c\ne0$$

• $|c|>1$ gives a horizontal compression by factor $\frac{1}{|c|}$
• $0<|c|<1$ gives a horizontal stretch by factor $\frac{1}{|c|}$
• $c<0$ also reflects across the $y$-axis

Examples:

• $3^{2x}=9^x$
• $16^{x/2}=4^x$
• $5^{-2x}=\left(\frac{1}{25}\right)^x$

That reflection idea is worth seeing directly:

$$b^{-x}=\frac{1}{b^x}=\left(\frac1b\right)^x$$

So changing $x$ to $-x$ flips the graph across the $y$-axis and swaps the base to its reciprocal.

How to Rewrite Complicated Expressions Efficiently

There’s a clean pattern to these.

1. Split exponent sums or differences.
2. Pull constant pieces out to become coefficients.
3. Rewrite variable multiples as a changed base.
4. Evaluate rational powers with roots.

Model example:

$$8^{(2x-3)/3} \to 8^{2x/3-1} \to 8^{2x/3}\cdot 8^{-1} \to (8^{2/3})^x\cdot \frac18 \to \frac18 4^x$$

That final form makes graph behavior much easier to read.

Common Mistakes and Fast Checks

The usual mistakes are very fixable:

• $b^m b^n$ means add exponents, but $(b^m)^n$ means multiply
• $2^x\cdot 3^x=6^x$, but $2^x\cdot 3^y$ does not combine that way
• $b^{-n}\neq -b^n$
• $x+k$ shifts left
• In $b^{cx}$, the horizontal factor is $\frac1{|c|}$, not $|c|$
• $b^m+b^n$ does not simplify to $b^{m+n}$

Fast check for equivalence. Plug in an easy value like $x=0$. If both forms match there and your algebra was legal, you’re probably on track.