Solve exponential equations using the properties of exponents (rewriting with a common base), and distinguish between situations that can be modeled with linear functions and with exponential functions (TEKS A.9D, A.9G).

What this topic is asking

TEKS A.9D and A.9G round out the Exponential Functions category with two skills: solving a simple exponential equation by rewriting both sides with a common base, and distinguishing linear from exponential situations. The second is a recurring conceptual item: linear means a constant difference, exponential a constant ratio.

Solving by a common base

When both sides of an exponential equation can be written with the same base, the exponents must match.

For $3^{2x} = 81$: write $81 = 3^4$, so $3^{2x} = 3^4$, giving $2x = 4$ and $x = 2$. The skill is recognizing the common base: $4, 8, 16, 32$ are powers of 2; $9, 27, 81$ are powers of 3.

Linear versus exponential

This is a high-frequency conceptual item. Look at how the outputs change for equal steps in the input:

• Constant difference: the outputs go up (or down) by the same amount each step. This is linear, $y = mx + b$, where the difference is the slope.
• Constant ratio: the outputs are multiplied by the same factor each step. This is exponential, $y = ab^x$, where the ratio is the base.

For the table $4, 12, 36, 108$: differences $8, 24, 72$ are not constant, but ratios $3, 3, 3$ are, so it is exponential with base 3.

Why the distinction matters

Linear and exponential models behave very differently over time: a linear quantity grows by a fixed amount, but an exponential quantity eventually outpaces any linear one because it multiplies. Recognizing which model fits a description, "increases by $50 each month" (linear) versus "increases by 5$y = mx + b$or$y = ab^x$.

How STAAR examines this topic

• Multiple choice. Solve a common-base equation, with the "forgot the constant in the exponent" distractor.
• Inline choice. Classify a table or description as linear or exponential, and pick the matching function.
• Equation editor and number entry. Enter the solution to an exponential equation.

A clarifying idea is that "by a fixed amount" is linear and "by a fixed percent or factor" is exponential: the wording of a problem usually signals the model before you do any algebra, and a table confirms it through constant difference versus constant ratio.

Recognizing common powers quickly

The whole common-base method depends on spotting that a number is a power of a small base, so it pays to know the short tables cold. Powers of 2 are $2, 4, 8, 16, 32, 64$; powers of 3 are $3, 9, 27, 81$; powers of 5 are $5, 25, 125$; and powers of 4 ($4, 16, 64$) and 9 ($9, 81$) overlap with powers of 2 and 3 respectively. When an equation pairs a base with a number on the other side, ask "is that number a power of this base?" If $2^{x} = 64$, then $64 = 2^6$ gives $x = 6$ at once. The few cases STAAR uses are always built from these small powers, so recognition is fast once the tables are automatic.

Where the two models cross

Comparing a linear and an exponential model shows why the distinction matters in practice. Suppose one savings plan adds \$100 a year (linear) and another grows 10$100 beats a small percent of a small balance, but the exponential plan's growth accelerates and eventually overtakes it permanently. This crossover is the intuition behind A.9G: the question of which model fits is not just bookkeeping, it changes the long-run behavior, and an item may ask you to identify which description (fixed amount or fixed percent) produces the faster eventual growth.

Try this

Q1. Solve $2^{x} = 32$. [1 point]

• Cue. $32 = 2^5$, so $x = 5$.

Q2. A table shows $5, 10, 15, 20$ for inputs $0, 1, 2, 3$. Linear or exponential? [1 point]

• Cue. Constant difference of 5: linear.