# How to solve this logarythmic inequality ?

• berdan
In summary, a logarithmic inequality is an inequality that contains a variable inside a logarithm function. To solve a logarithmic inequality, you need to isolate the logarithm on one side of the inequality and then exponentiate both sides using the appropriate base. The process for solving a logarithmic equation and a logarithmic inequality are different, with the latter involving finding the range of values for the variable that satisfy the inequality. You can use a calculator to solve logarithmic inequalities, but it is important to understand the manual process first. Some common mistakes to avoid when solving logarithmic inequalities include forgetting to check for extraneous solutions, using the wrong base when exponentiating, and not accounting for restrictions on the variable. It is important to carefully follow the

## Homework Statement

Here is the problem :
https://www.dropbox.com/s/otzzne7wjyuqa5o/question.jpg?dl=0
I've tried solving this inequality but alas,nothing...It's an exam question for my student,and for my great shame,I have no idea how to solve it :(

What have you tried so far? I know it looks scary, but why don't you try using some of your laws of logarithms.

You might also want to use the fact that
$$log_a(x)= \frac{log_b(x)}{log_b(a)}$$
to get the two logarithms to the same base.

## What is a logarithmic inequality?

A logarithmic inequality is an inequality that contains a variable inside a logarithm function. For example, log(x) > 2 is a logarithmic inequality.

## How do I solve a logarithmic inequality?

To solve a logarithmic inequality, you need to isolate the logarithm on one side of the inequality and then exponentiate both sides using the appropriate base. You may also need to check for extraneous solutions and account for any restrictions on the variable.

## What is the difference between solving a logarithmic equation and a logarithmic inequality?

The process for solving a logarithmic equation involves finding the value of the variable that makes the equation true. On the other hand, solving a logarithmic inequality involves finding the range of values for the variable that satisfy the inequality.

## Can I use a calculator to solve logarithmic inequalities?

Yes, you can use a calculator to solve logarithmic inequalities. However, it is important to understand the steps involved in solving them manually before relying on a calculator.

## Are there any common mistakes to avoid when solving logarithmic inequalities?

Yes, some common mistakes to avoid when solving logarithmic inequalities include forgetting to check for extraneous solutions, using the wrong base when exponentiating, and not accounting for restrictions on the variable. It is important to carefully follow the steps and double-check your work to avoid these mistakes.