Exponential inequalities

An exponential inequality is an inequality where the variable appears in the exponent, such as $2^x > 8$ or $3^{x+1} \leq 27$. To solve these, we often use properties of exponents and compare powers with the same base.

Important properties

• If $a > 1$, then $a^x$ increases as $x$ increases (it is strictly increasing).

• If $0 < a < 1$, then $a^x$ decreases as $x$ increases (it is strictly decreasing).

• To compare $a^x$ and $a^y$ for $a > 0$, consider the sign of $x - y$ and whether $a > 1$ or $0 < a < 1$.

• Sometimes, taking logarithms on both sides helps to solve the inequality.