Exponential inequalities are inequalities in which the unknown appears in the exponent. They represent one of the fundamental applications of the properties of exponential functions and require careful attention to monotonicity.
The simplest form is:
\[ a^{f(x)} \gtrless a^{g(x)}, \qquad a>0,\ a\neq 1. \]
In these cases, the behavior of the inequality depends entirely on the base \(a\):
- if \(a>1\), the exponential function is strictly increasing;
- if \(0<a<1\), the exponential function is strictly decreasing.
Consequently:
\[ a^{u}>a^{v} \iff u>v \qquad \text{if } a>1, \]
whereas:
\[ a^{u}>a^{v} \iff u<v \qquad \text{if } 0<a<1. \]
This is the central principle underlying the entire theory of exponential inequalities.
Table of Contents
- Definition of an exponential inequality
- Monotonicity of the exponential function
- Basic inequalities with the same base
- Case \(a>1\)
- Case \(0<a<1\)
- Reducing to a common base
- Inequalities reducible to a single exponential
- The substitution method
- Rational exponential inequalities
- Systems of exponential inequalities
- Practice Problems
Definition of an exponential inequality
An exponential inequality is an inequality in which the unknown appears in the exponent of at least one power.
Examples:
\[ 2^x>8, \]
\[ 3^{2x-1}\le 9, \]
\[ \left(\frac12\right)^{x+1}>4. \]
Not all exponential inequalities are solved in the same way. In some cases it is enough to compare the exponents directly; in others, algebraic manipulations, factoring, or substitutions are required.
Monotonicity of the exponential function
Consider the function:
\[ f(x)=a^x, \qquad a>0,\ a\neq 1. \]
It is:
- increasing if \(a>1\);
- decreasing if \(0<a<1\).
This fact is fundamental because it allows us to pass from an exponential inequality to an inequality between the exponents.
Indeed:
\[ a^{u(x)} \gtrless a^{v(x)} \]
is equivalent to:
\[ u(x)\gtrless v(x) \]
when \(a>1\), whereas the direction of the inequality reverses when \(0<a<1\).
Basic inequalities with the same base
Consider:
\[ 5^{2x-1}>5^3. \]
Since the base is greater than \(1\), we can compare the exponents directly:
\[ 2x-1>3. \]
Solving:
\[ 2x>4 \]
\[ x>2. \]
Therefore:
\[ S=(2,+\infty). \]
Case \(a>1\)
When the base is greater than \(1\), the exponential function preserves the order of the inequality:
\[ a^u>a^v \iff u>v. \]
Example:
\[ 3^{x+2}\le 3^{2x-1}. \]
Comparing the exponents:
\[ x+2\le 2x-1. \]
Hence:
\[ 3\le x. \]
The solution set is:
\[ S=[3,+\infty). \]
Case \(0<a<1\)
When:
\[ 0<a<1, \]
the exponential function is decreasing, so the direction of the inequality is reversed.
For example:
\[ \left(\frac12\right)^{x-1}>\left(\frac12\right)^{2x+3}. \]
Since:
\[ 0<\frac12<1, \]
we reverse the inequality sign:
\[ x-1<2x+3. \]
Thus:
\[ -4<x. \]
Therefore:
\[ S=(-4,+\infty). \]
Reducing to a common base
Often the bases are different but can be expressed as powers of a common base.
Consider:
\[ 8^x>2^{x+1}. \]
Observe that:
\[ 8=2^3. \]
Then:
\[ (2^3)^x>2^{x+1}. \]
Applying the power rule:
\[ (a^m)^n=a^{mn}, \]
we obtain:
\[ 2^{3x}>2^{x+1}. \]
Since \(2>1\):
\[ 3x>x+1. \]
Hence:
\[ 2x>1 \]
\[ x>\frac12. \]
Inequalities reducible to a single exponential
Sometimes the expression must be simplified before monotonicity can be applied.
For example:
\[ 2^{x+1}-2^x>4. \]
Factoring out \(2^x\):
\[ 2^x(2-1)>4, \]
that is:
\[ 2^x>4. \]
Since:
\[ 4=2^2, \]
we get:
\[ 2^x>2^2. \]
Therefore:
\[ x>2. \]
The substitution method
Some exponential inequalities take on a polynomial form after a suitable substitution.
Consider:
\[ 2^{2x}-5\cdot 2^x+6>0. \]
Let:
\[ t=2^x. \]
Since an exponential function is always positive:
\[ t>0. \]
Also:
\[ 2^{2x}=(2^x)^2=t^2. \]
The inequality becomes:
\[ t^2-5t+6>0. \]
Factoring:
\[ (t-2)(t-3)>0. \]
A sign analysis gives:
\[ t<2 \quad \text{or} \quad t>3. \]
Substituting back:
\[ 2^x<2 \quad \text{or} \quad 2^x>3. \]
The first condition gives:
\[ x<1. \]
The second:
\[ x>\log_2 3. \]
Therefore:
\[ S=(-\infty,1)\cup(\log_2 3,+\infty). \]
Rational exponential inequalities
Rational expressions involving exponentials may also arise.
Example:
\[ \frac{2^x-1}{2^x+3}>0. \]
Let:
\[ t=2^x, \qquad t>0. \]
We obtain:
\[ \frac{t-1}{t+3}>0. \]
Since:
\[ t+3>0 \]
for all \(t>0\), it suffices to require:
\[ t-1>0. \]
So:
\[ t>1. \]
Returning to the original variable:
\[ 2^x>1. \]
Since:
\[ 1=2^0, \]
we conclude:
\[ x>0. \]
Systems of exponential inequalities
Exponential inequalities may also appear within systems.
For example:
\[ \begin{cases} 2^x>4 \\ 3^x\le 27 \end{cases} \]
The first inequality gives:
\[ x>2. \]
The second:
\[ x\le 3. \]
Taking the intersection:
\[ S=(2,3]. \]
Practice Problems
Example 1
Solve:
\[ 4^x\ge 16. \]
Write everything in base \(2\):
\[ 4=2^2, \qquad 16=2^4. \]
Then:
\[ 2^{2x}\ge 2^4. \]
Since \(2>1\):
\[ 2x\ge 4. \]
Thus:
\[ x\ge 2. \]
Therefore:
\[ S=[2,+\infty). \]
Example 2
Solve:
\[ \left(\frac13\right)^{2x-1}<27. \]
Write everything in base \(3\):
\[ \left(\frac13\right)^{2x-1}=3^{-(2x-1)}, \qquad 27=3^3. \]
This gives:
\[ 3^{-2x+1}<3^3. \]
Since the base \(3\) is greater than \(1\):
\[ -2x+1<3. \]
Hence:
\[ -2x<2 \]
\[ x>-1. \]
Therefore:
\[ S=(-1,+\infty). \]
Example 3
Solve:
\[ 3^{2x}-10\cdot 3^x+9\le 0. \]
Let:
\[ t=3^x, \qquad t>0. \]
We obtain:
\[ t^2-10t+9\le 0. \]
Factoring:
\[ (t-1)(t-9)\le 0. \]
From the sign analysis:
\[ 1\le t\le 9. \]
Returning to the exponential:
\[ 1\le 3^x\le 9, \]
that is:
\[ 3^0\le 3^x\le 3^2. \]
Since \(3>1\):
\[ 0\le x\le 2. \]
Therefore:
\[ S=[0,2]. \]
Exponential inequalities are solved by exploiting the fundamental properties of the exponential function: monotonicity, comparison of bases, algebraic manipulation, and substitution. Understanding how the base governs the behavior of the function is the essential step — it is precisely what determines whether the direction of the inequality is preserved or reversed, and what makes for a rigorous and correct solution.