Exponential inequalities are inequalities that involve exponential expressions. Solving these can seem tricky at first, but with a few steps, you can master them.
Step-by-Step Process
- Understand the Problem
Consider the inequality $2^x > 8$. Here, you need to find the values of $x$ that make the inequality true.
- Rewrite the Inequality with the Same Base
Rewrite both sides of the inequality using the same base if possible. For example, $8$ can be written as $2^3$. So, $2^x > 8$ becomes $2^x > 2^3$
- Compare the Exponents
Once the bases are the same, you can compare the exponents directly. Therefore, $2^x > 2^3$ simplifies to $x > 3$
- Check Your Solution
Always verify your solution by plugging it back into the original inequality. For $x > 3$, if $x = 4$, then $2^4 = 16$ which is indeed greater than $8$. So, $x > 3$ is correct.
Example Problems
Example 1: $3^{x+1} leq 27$
- Rewrite $27$ as $3^3$: $3^{x+1} leq 3^3$
- Compare the exponents: $x+1 leq 3$
- Solve for $x$: $x leq 2$
Example 2: $5^{2x} < 125$
- Rewrite $125$ as $5^3$: $5^{2x} < 5^3$
- Compare the exponents: $2x < 3$
- Solve for $x$: $x < frac{3}{2}$
Special Cases
Different Bases
If the bases are different and cannot be rewritten to match, you may need to use logarithms. For example, to solve $2^x > 3$, take the logarithm of both sides:
$log(2^x) > log(3)$
Using the property of logarithms, this becomes:
$x log(2) > log(3)$
Solve for $x$:
$x > frac{log(3)}{log(2)}$
Negative Exponents
When dealing with negative exponents, remember that $a^{-x} = frac{1}{a^x}$. For example, $2^{-x} < 4$ becomes $frac{1}{2^x} < 4$. Rewriting $4$ as $2^2$, we get $frac{1}{2^x} < 2^2$. Invert both sides to get $2^x > frac{1}{2^2}$, which simplifies to $2^x > 0.25$. Solve as usual.
Conclusion
Solving exponential inequalities involves rewriting the expressions to have the same base, comparing the exponents, and verifying the solutions. When the bases are different, logarithms can be used. Understanding these steps will help you solve any exponential inequality you encounter.