Solve Logarithmic Inequality: log₀.₂₅7 + log₀.₂₅(1/3) < log₀.₂₅(x²)

Q: Why does the inequality flip when the base is less than 1?

When the base is between 0 and 1, the logarithmic function is decreasing. This means larger inputs give smaller outputs! So if $ \log_{0.25}(a) , then \( a > b $.

Q: How do I handle the domain restriction x² in the logarithm?

Since we need $ x^2 > 0 $, this means x ≠ 0. The solution \( -\sqrt{\frac{7}{3}} must exclude x = 0, giving us two intervals.

Q: Can I solve this without combining the logarithms first?

You could, but it's much harder! Using the property $ \log_b(a) + \log_b(c) = \log_b(ac) $ simplifies the left side to one logarithm, making the inequality easier to solve.

Q: What if I forgot that 0.25 = 1/4?

That's okay! You can work with 0.25 directly. The key insight is that 0.25 , so you still need to flip the inequality. Converting to $ \frac{1}{4} $ just makes some students more comfortable.

Q: Why isn't the answer just one continuous interval?

Because of the domain restriction! We need $ x^2 > 0 $, which means x ≠ 0. So our solution \( |x| splits into two parts: negative values and positive values.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:05 This is the domain of definition

00:10 We'll use the formula for logical addition, we'll get the log of their product

00:25 Let's compare the numbers

00:35 Let's find the appropriate domain

00:50 Let's check the domain of definition to find the solution

00:55 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

$\log_{0.25}7+\log_{0.25}\frac{1}{3}<\log_{0.25}x^2$

$x=\text{?}$

2

Step-by-step solution

Let's solve the inequality step-by-step:

Step 1: Apply the sum of logarithms property.
We have:

\log_{0.25}\left(7 \times \frac{1}{3}\right) < \log_{0.25}(x^2)

This simplifies to:

\log_{0.25}\left(\frac{7}{3}\right) < \log_{0.25}(x^2)

Step 2: Use the property of logarithms indicating that if bases are the same and the inequality involves $\log_b(a) < \log_b(b)$ , where $b < 1$ , it implies:

a > b

Since $0.25 < 1$ , the inequality $\log_{0.25}\left(\frac{7}{3}\right) < \log_{0.25}(x^2)$ implies:

\frac{7}{3} > x^2

Step 3: Simplify the inequality:
$x^2 < \frac{7}{3}$

Since $x^2 < \frac{7}{3}$ , this implies:

-\sqrt{\frac{7}{3}} < x < \sqrt{\frac{7}{3}}

Thus, the domain of $x$ based on the restriction of positive numbers for logarithm and quadratic expression is:

-\sqrt{\frac{7}{3}} < x < 0 \text{ and } 0 < x < \sqrt{\frac{7}{3}}

Therefore, the correct solution is $-\sqrt{\frac{7}{3}} < x < 0, 0 < x < \sqrt{\frac{7}{3}}$ .

Thus, the choice that corresponds to this solution is Choice 1.

3

Final Answer

$-\sqrt{\frac{7}{3}} < x < 0,0 < x < \sqrt{\frac{7}{3}}$

Key Points to Remember

Essential concepts to master this topic

Base Property: When base < 1, inequality direction reverses after removing log
Technique: Combine logs first: log₀.₂₅(7) + log₀.₂₅(1/3) = log₀.₂₅(7/3)
Check: Test x = 1: log₀.₂₅(7/3) < log₀.₂₅(1), since 7/3 > 1 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to reverse inequality sign with base less than 1
Don't treat log₀.₂₅(a) < log₀.₂₅(b) like log₂(a) < log₂(b) = wrong direction! When the base is between 0 and 1, the logarithmic function is decreasing, so the inequality flips. Always remember: base < 1 means flip the inequality sign.

Practice Quiz

Test your knowledge with interactive questions

$\log_{10}3+\log_{10}4=$

FAQ

Everything you need to know about this question

Why does the inequality flip when the base is less than 1?

+

When the base is between 0 and 1, the logarithmic function is decreasing. This means larger inputs give smaller outputs! So if $\log_{0.25}(a) < \log_{0.25}(b)$ , then $a > b$ .

How do I handle the domain restriction x² in the logarithm?

+

Since we need $x^2 > 0$ , this means x ≠ 0. The solution $-\sqrt{\frac{7}{3}} < x < \sqrt{\frac{7}{3}}$ must exclude x = 0, giving us two intervals.

Can I solve this without combining the logarithms first?

+

You could, but it's much harder! Using the property $\log_b(a) + \log_b(c) = \log_b(ac)$ simplifies the left side to one logarithm, making the inequality easier to solve.

What if I forgot that 0.25 = 1/4?

+

That's okay! You can work with 0.25 directly. The key insight is that 0.25 < 1, so you still need to flip the inequality. Converting to $\frac{1}{4}$ just makes some students more comfortable.

Why isn't the answer just one continuous interval?

+

Because of the domain restriction! We need $x^2 > 0$ , which means x ≠ 0. So our solution $|x| < \sqrt{\frac{7}{3}}$ splits into two parts: negative values and positive values.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rules of Logarithms questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Solve Logarithmic Inequality: log₀.₂₅7 + log₀.₂₅(1/3) < log₀.₂₅(x²)

Logarithmic Inequalities with Base Between 0 and 1

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why does the inequality flip when the base is less than 1?

How do I handle the domain restriction x² in the logarithm?

Can I solve this without combining the logarithms first?

What if I forgot that 0.25 = 1/4?

Why isn't the answer just one continuous interval?

🌟 Unlock Your Math Potential

More Questions

Rules of Logarithms

Solve Base-13 Logarithmic Inequality: log₁₃(2x² + 3) ≤ log₁₃7 - log₁₃x²

Solve the Logarithmic Inequality: log₃(5x) × log₁/₇(9) ≥ log₁/₇(4)

Solve the Logarithmic Inequality: Determine x in log_4(7) + log_4(2) ≤ log_4(x)

Solve Complex Logarithmic Equation: 1/(log_x3) × x²log_(1/x)27 + 4x + 6 = 0

Rules of Logarithms Combined

Solve the Logarithmic Inequality: log₁/₂(5) - log₁/₂(4) ≤ log₁/₂(x) - log₁/₂(3)

Solve the Logarithmic Inequality: log₂3 - log₂(x+3) ≤ 8

Solve: Complex Logarithmic Fraction with log₄7 and log₄₉a Terms

Solve the Complex Logarithmic Equation: (2ln4/ln5) + 1/log₅ = log₅(7x²+9x)

The Sum of Logarithms

Solve log₁₀3 + log₁₀4: Step-by-Step Logarithm Addition

Solve log₂4 + log₂5: Adding Logarithms with Base 2

Solve the Logarithmic Inequality: ln(x+5) + ln(x) ≤ ln(4) + ln(2x)

Simplify: -3(ln4/ln5 - log₅7 + 1/log₆5) Logarithmic Expression