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 loga.

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 logax > logb x ln a > ln b xloga > logb x > ln b ln a x > logb loga

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 logax > logb x ln a > ln b xloga > logb x < ln b ln a x < logb loga

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 | ÷log0.25 x < lg 9 log0.25 1.6

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