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CAT Solving logarithm inequalities - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Syllabus

Quick Facts

4 Questions around this concept.

Solve by difficulty

EASY

Solution of $\log _{5}(x-2) \geqslant \log _{5}(2 x)$

$\left ( -\infty ,-2 \right )$

$\left ( 3, \infty \right )$

$\left ( 0, \infty \right )$

$\phi$

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Concepts Covered - 1

Solving logarithm inequalities

Logarithm inequalities are algebraic inequalities that involve logarithmic functions. Solving these inequalities requires a solid understanding of the properties of logarithms and their behaviour. Just like linear inequalities, the goal is to isolate the variable and determine its range of possible values.

Basics of Logarithm Inequalities:

1. Domain of Logarithmic Functions: The argument of the logarithm (the number inside the logarithm) must always be positive. This is crucial when considering the domain of solutions.

2. Monotonicity: For any base b > 1, the logarithm function is increasing. However, for 0 < b < 1, the logarithm function is decreasing.

Solved Examples:

1. Solve for $\mathrm{x: \log _{10}(x+3)>1}$

Solution:

Convert the logarithmic inequality to its exponential form:

$\mathrm{ x + 3 > 10^{1}}$

x + 3 > 10

x > 7.

2. Solve for $\mathrm{x: 2 \log _3(x)+\log _3(x-2)<3 .}$ .

Solution:

Using properties of logarithms:

$\mathrm{\log _3\left(x^2\right)+\log _3(x-2)<3}$

Combining logs:

$\mathrm{\log _3\left(x^3-2 x^2\right)<3}$

Convert to exponential form:

$\begin{gathered} \\x^3-2 x^2<27 \\ x^3-2 x^2-27<0 \end{gathered}$

This cubic inequality can be solved using factorization or graphical methods.

Tips and Tricks:

1. Use Properties: Utilise logarithm properties to simplify and combine terms. This often makes the inequality more manageable.

2. Watch the Base: Remember, if the base is between 0 and 1, the logarithm function is decreasing, which means the inequality sign will flip when dividing or multiplying by a negative value.

3. Domain Consideration: Always ensure that the argument of the logarithm is positive. This might restrict the range of possible solutions.

4. Graphical Insight: If you're familiar with the shape and behaviour of logarithmic graphs, they can offer a visual method to solve inequalities, especially when combined with other functions.

5. Exponential Conversion: When stuck, try converting the logarithmic inequality into its exponential form. This can sometimes simplify the problem or provide a new perspective.

6. Test Points: Once you have potential solution intervals, test points from each interval in the original inequality to ensure they're valid.

In conclusion, solving logarithm inequalities is about understanding the behaviour of logarithms and methodically applying algebraic techniques. With a solid grasp of logarithm properties and consistent practice, students will be adept at solving such inequalities, ensuring they're well-prepared for related questions in their entrance exams.