a^(u(x)) > a^(v(x)), a > 1
Same base, base greater than 1
Keep the inequality direction and solve u(x) > v(x).
2^(x + 1) > 8 becomes x + 1 > 3.
Exponential Inequality Calculator
Solve exponential inequalities like 2^(x+1) > 8 — see exactly when the inequality sign flips based on the base.
exponential inequality calculatorsolve exponential inequalitydifferent bases with logs
What this page explains
Same base: compare exponents, but flip the sign when 0 < base < 1.
Different bases: take ln on both sides before solving the linear inequality.
Every answer is paired with an exponential graph and interval notation.
Calculator
Solve exponential inequalities with the right path
Same-base problems compare exponents. Different-base problems use logs first.
Or paste a full expression
Current problem
Answer
Interval notation
1
Step 1 — Identify the shared base.
Rewrite the constant side as a power of base 0.5 before comparing exponents.
2
Step 2 — Base between 0 and 1 flips the sign.
Since 0 < 0.5 < 1, the exponential function is decreasing. Larger exponents make smaller values, so the inequality sign flips.
3
Step 3 — Solve the exponent inequality.
Subtract the right exponent from the left exponent, then isolate x.
4
Step 4 — Write the answer in interval notation.
The final boundary becomes the endpoint of the interval. Strict symbols use parentheses; inclusive symbols use brackets.
Exponential Inequality Graph
Where the curves cross becomes the interval endpoint
Shade: right of x = -2
Left curve
0.5^(x)
Right curve
0.5^(-2)
Solution
The solution is x ≥ -2.
Number Line
Drag to pan. Use the controls or mouse wheel to zoom the view.
Open circles exclude endpoints. Closed circles include them.Export the current graph as SVG or PNG.
Interactive Flip Rule
Drag the base and watch the inequality direction change
The only mystery is whether a^x is increasing or decreasing. When the base slides below 1, the curve falls as x grows, so same-base inequalities reverse.
Base is between 0 and 1 -> inequality FLIPS
0.5^3 < 0.5^1, even though 3 > 1. The curve is decreasing, so the value comparison points the other way.
Step-by-Step Breakdown
Two solving paths, chosen by the bases
Same base
If a > 1, keep the sign. If 0 < a < 1, the function decreases, so flip the sign before solving the exponent inequality.
Different bases
The exponents cannot be compared directly. Logs convert the problem into a linear inequality while preserving the comparison.
Method Selector
Pick the right exponential inequality path first
a^(u(x)) > a^(v(x)), 0 < a < 1
Same base, base between 0 and 1
Flip the inequality direction because the exponential function is decreasing.
(1/2)^x <= (1/2)^(-2) becomes x >= -2.
a^(u(x)) > b^(v(x))
Different bases
Use logarithms, collect x terms, then solve the resulting linear inequality.
3^x > 2^(x + 1) becomes x ln(3) > (x + 1) ln(2).
a^x compared with a negative number
Range-only cases
Use the fact that valid exponential outputs are always positive.
3^x > -7 is true for every real x.
Avoid These Mistakes
Rules that keep exponent answers valid
Comparing exponents when the bases are different
Only compare exponents directly after rewriting both sides with the same positive base.
Forgetting the fractional-base flip
When 0 < base < 1, the function decreases, so the inequality direction reverses.
Taking logs of a nonpositive side
Handle range cases first. If the right side is negative, many exponential comparisons are automatically true or false.
Examples
Common exponential inequality examples
Base greater than 1
2^(x+1) > 8
Rewrite 8 as 2^3, keep the sign because base 2 is increasing, then solve x + 1 > 3.
Base between 0 and 1
(1/2)^x <= 4
Rewrite 4 as (1/2)^(-2), then flip <= to >= because base 0.5 is decreasing.
Sign flipped because 0 < base < 1.
Always true range case
3^x > -7
Since 3^x is always positive, it is automatically greater than -7 for every real x.
FAQ
Frequently Asked Questions
How do you solve an exponential inequality?
First try to rewrite both sides with the same base. Then compare the exponents, keeping the inequality direction when the base is greater than 1 and flipping it when the base is between 0 and 1.
Why does the inequality sign flip when the base is less than 1?
If 0 < a < 1, the function a^x is decreasing. Larger exponents create smaller values, so a comparison such as (1/2)^x <= (1/2)^(-2) becomes x >= -2.
What is the difference between exponential equations and exponential inequalities?
An exponential equation looks for exact crossing points where two expressions are equal. An exponential inequality asks where one expression is greater or less, so the final answer is usually an interval.
Can an exponential inequality have no solution?
Yes. Because a^x is always positive for valid real exponential bases, an inequality such as 2^x < 0 has no real solution. An inequality such as 2^x > -5 is true for all real numbers.
How do you solve exponential inequalities with different bases?
If the bases cannot be matched as powers of one common base, take natural logs on both sides, use the power rule, collect x terms, and solve the resulting linear inequality.
What if the exponential inequality can't be rewritten with the same base?
Use logarithms. This page shows the log-conversion path for different-base inputs, and the logarithmic inequality calculator covers the related log-specific rules.
Can you compare exponents when the bases are different?
No. Direct exponent comparison only works after both sides are rewritten with the same base. If the bases cannot be matched, use logarithms or a graph-supported numeric comparison.
Why are some exponential inequalities true for all real numbers?
For valid positive bases other than 1, a^x is always positive. That means an inequality such as 3^x > -7 is true for all real x, while 3^x < -7 has no real solution.