Last post, we talked about radical inequalities. In this post, we will talk about how to solve logarithmic inequalities. We’ll see logarithmic inequalities in forms such as or . In order to solve these inequalities, the goal will be to isolate the variable, just as in any inequality, and we will do this by getting rid of the log function. Let’s dive in and see how to solve logarithmic inequalities.

Steps to solve logarithmic inequalities:

1. Use algebraic manipulation to move anything that is not in the logarithmic expression to one side

2. Combine logarithmic expressions

3. Isolate the variable by getting rid of the logarithmic expression

Ex:

Ex:

4. Solve inequality

5. Get the range for the expression in the original log function

Ex:

Range:

6. Combine ranges

Let’s do an example step by step now.

First example (click
here):

Step 1: Use algebraic manipulation to move anything that is not in the logarithmic expression to one side

There’s nothing to move, so we can skip this step.

Step 2: Combine logarithmic expressions

Step 3: Isolate the variable by getting rid of the logarithmic expression

Step 4: Solve inequality

We can see that this is now a rational inequality. We won’t solve this step by step; I will show the answer after solving this inequality. If you are struggling with solving this inequality, visit the blog post on rational inequalities.

Step 5: Get the range for the expression in the original log function

Step 6: Combine ranges

, ,

That wasn’t too bad! Let’s see some more examples.

Second example (click
here):

Last example (click
here):

Solving logarithmic inequalities is not too difficult. Just remember to get the ranges inside the logarithmic expressions and to double check your work. For more help and practice on this topic visit Symbolab’s practice.

Until next time,

Leah

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