Solving Exponential and Logarithmic Inequalities

In this lesson, we will learn about solving exponential and logarithmic inequalities. When faced with an exponential inequality, one approach is to rewrite the inequality using the same base for both sides and then set up an inequality with the exponents. This works well in the case where both sides can easily be written with the same base. There is a bit of a rule to consider. When a^x > a^y and a > 1, we can just say x > y and solve the inequality. When 0 < a < 1, then the direction of the inequality symbol must be flipped. If we can't easily write each side with the same base, we can turn to logarithms to solve our problem. For these problems, we must consider the rule when taking logarithms of each side. When a > 1 and x > y, then log_a(x) > log_a(y). When 0 < a < 1, we have to flip the direction of the inequality symbol. When solving logarithmic inequalities, we generally use the test-point approach. We first find the domain. Then we will solve the related equation by replacing the inequality symbol with an equality symbol. Lastly, we will set up intervals on the number line and then test to obtain our final solution.