MathInMed

Please Wait...

Logarithmic Inequalities

Welcome!

I. What can I expect?

Four types of logarithmic inequalities
- comparison of a logarithm and a number

- comparison of two logarithms

- comparison of a logarithmic and a non-logarithmic function

- composition of functions

- logarithms with different basis

- problems, which just use logarithmic inequalities
Challenging questions
Different solutions
Deep explanation of used procedures

Logarithmic Inequalities - content MAP

II. What shoul I do?

THINK, THINK, THINK!

Think about everything what you read/watch! That is the only way how to build the knowledge :)

CHALLENGE YOURSELF

Always try to understand! If there is something you do not understand, think about it first. Later, search for the explanation.

CHALLENGE EVERY PROCEDURE

Always ask yourself, does this procedure work also under changed conditions??? If no, why?

III. What is my reward?

As a student...

Great feeling as a reward for your hard work :)

Deep understanding of used procedures

Ability to solve (ideally :)) every logarithmic inequality from different textbooks

As a teacher...

Experience of constructivist teaching and learning

Ability to motivate your students by raising constructive questions

Content map and categorization of problems related to Logarithmic Inequalities

IV. How should I start?

Follow the content map below and choose problem you are interested in. If you do not know, start with the first one from the first group :)

Logarithmic Inequalities

Group I
$$\log_{a}x < b$$

$$\log_{\frac{1}{4}}x < 1 $$

$$\log_{5}0.5 \leq \log_{5}x $$

$$\log_{e}x \geq -2 $$

$$ 0 > 5 + \log_{5}x $$

$$3\log_{\frac{1}{2}}x > \scriptsize \frac{1}{4}$$

$$-\log_{\frac{3}{2}}0.5 \leq \log_{\frac{2}{3}}\pi$$

Group II
$$\log_{a}f(x) < \log_{a}g(x) $$

$$\log2x < \log3x $$

$$\log_{0.5}(x^2-9) \geq 0 $$

$$\log_{x}10 \leq -1 $$

Group III
$$\log_{a}f(x) < \log_{b}g(x) + h(x)$$

$$\log_3(x^2 - 4) > 1 + \log_2(-x)$$

$$\log_{\frac{1}{2}}x^2 \geq \log_{\frac{1}{2}}(3 - x) -1 $$

$$\log_{\frac{1}{3}}(log^2|x|-3\log|x| + 5) \leq 2 $$

Group IV
Logarithm in context

Find the Domain of a Function: $$f: y = \sqrt{\log_{0.1}(x - 2)}$$

Find parameter a. Following equation
has to have at least one root: $$x^2 - 4x - \log_{a}x = 0$$

Let's start: problem I