Advanced Math / Nonlinear Equations One Variable
Exponential Equations in One Variable
Exponential equations on the SAT often become linear equations after rewriting both sides with the same base. When bases match, the exponents can be set equal.
Least you need to know
- If $a^m=a^n$ for $a>0$ and $a\ne1$, then $m=n$.
- Numbers such as 4, 8, 16, 32, and 64 can be rewritten as powers of 2.
- Numbers such as 9, 27, and 81 can be rewritten as powers of 3.
- A negative exponent represents a reciprocal, such as $3^{-3}=\frac{1}{27}$.
Key notation
- $2^{x+3}=4^x$ — An exponential equation that can be solved by rewriting $4^x$ as $2^{2x}$
- $3^x=\frac{1}{27}$ — An equation involving a negative exponent
Worked example
- To solve $2^{x+3}=4^x$, rewrite $4^x$ as $(2^2)^x=2^{2x}$.
- Then $2^{x+3}=2^{2x}$, so $x+3=2x$.
- Therefore, $x=3$.
Common mistakes
- Rewrite both sides using the same base before setting exponents equal.
- Do not ignore an added or subtracted constant in the exponent.
- A reciprocal such as $\frac{1}{27}$ is $3^{-3}$.
How to recognize it
- The problem asks for the solution of an equation with a variable in an exponent.
- Both sides can be written using the same base.
- The problem gives a parameter in the exponent and a known solution.
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