Exponential Equations on the SAT
Match Bases and Solve
Build equations from context, spot patterns fast, and practice with intent.
Why the SAT Emphasizes Exponential Equations
Exponential equations describe growth and decay, like population changes or compound interest. On the SAT, the bases are usually chosen so you can solve without logarithms, as long as you recognize the patterns.
This lesson shows how to rewrite both sides with the same base, how to compare exponents, and when it is faster to let Desmos find the intersection for you. You will also learn how to spot equations that look exponential but are actually linear after a rewrite.
A Simple Definition Unlocks Exponential Equations
If you can rewrite both sides with the same base, the equation becomes simple. For example, $27 = 3^3$, so $3^{x + 1} = 27$ means $3^{x + 1} = 3^3$ and therefore $x + 1 = 3$.
When the bases do not match cleanly, graphing both sides in Desmos is usually faster and less error prone. Always check that the base is positive and not equal to $1$, because those cases behave differently.
Work Through Exponential Equations Step by Step
Solve $3^{x + 1} = 27$.
Solve $3^{x + 1} = 27$ by rewriting $27$ as a power of $3$.
Rewrite $27$ as a power of $3$
Set the exponents equal once the bases match.
Solve for $x$ to isolate the variable.
Use Desmos to Check Exponential Equations
Solve $3^{x + 1} = 27$.
Desmos can solve exponentials by graphing both sides and finding the intersection. This is fast when bases do not match cleanly.
y = 3^(x + 1)
y = 27
Algebra is faster when bases match cleanly. Desmos is faster when you want a decimal approximation or a quick check.
Expert move: Graph the left and right sides directly (no need to isolate $y$), then click the intersection(s) and read the $x$-value(s). If Desmos gives decimals, convert to fractions when the choices are exact and keep only values that fit the domain or context.
When to skip Desmos: If the algebra is one or two steps, solve by hand and use Desmos only to verify; Desmos is best for messy coefficients or checking setup.
- Desmos features used: graphing, intersections.
- Common mistake: confusing the base with the exponent.
Practice Exponential Equations with SAT-Style Questions
Solve each exponential equation or model.
Solve .
Solve .
If , what is the growth factor each period?
Solve .
Key Takeaways to Remember for Exponential Equations
- Rewrite both sides with the same base to solve quickly.
- The base is the growth or decay factor in exponential models.
- Desmos is a good check when the base is not obvious.
About the Instructor
