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How to Solve Exponential Inequalities

When solving exponential inequalities it is very important that you look at the value of the argument a in the logarithm. That value determines whether you have to turn the inequality sign when you multiply or divide by ln ⁡ a or log ⁡ a.

Rule

Solving exponential inequalities

When a > 1, ln ⁡ (a) > 0 and you can solve the inequality as usual.

ax > b ax > b ln ⁡ ax > ln ⁡ b log ⁡ ax > log ⁡ b x ln ⁡ a > ln ⁡ b x log ⁡ a > log ⁡ b x > ln ⁡ b ln ⁡ a x > log ⁡ b log ⁡ a

When 0 < a < 1, ln ⁡ (a) < 0, you will have to turn the inequality sign as you end up dividing or multiplying by a negative number!

ax > b ax > b ln ⁡ ax > ln ⁡ b log ⁡ ax > log ⁡ b x ln ⁡ a > ln ⁡ b x log ⁡ a > log ⁡ b x < ln ⁡ b ln ⁡ a x < log ⁡ b log ⁡ a

Example 1

Solve the inequality 3.5x > 439

3.5x > 439 ln ⁡ 3.5x > ln ⁡ 439 x ln ⁡ 3.5 > ln ⁡ 439 | ÷ ln ⁡ 3.5 x > ln ⁡ 439 ln ⁡ 3.5 ≈4.9

Example 2

Solve the inequality 50 ⋅1.05x > 300

50 ⋅1.05x > 300 | ÷ 50 1.05x > 6 ln ⁡ 1.05x > ln ⁡ 6 x ln ⁡ 1.05 > ln ⁡ 6 | ÷ ln ⁡ 1.05 x > ln ⁡ 6 ln ⁡ 1.05 ≈36.7

Example 3

Solve the inequality 3 ⋅0.25x > 27

3 ⋅0.25x > 27 | ÷3 0.25x > 9 lg ⁡ 0.25x > lg ⁡ 9 xlg ⁡ 0.25 > lg ⁡ 9 | ÷log ⁡ 0.25 x < lg ⁡ 9 log ⁡ 0.25 ≈−1.6

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