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

log⁡47+log⁡42≤log⁡4x \log_47+\log_42\le\log_4x

x=? x=\text{?}

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

log⁡47+log⁡42≤log⁡4x \log_47+\log_42\le\log_4x

x=? x=\text{?}

2

Step-by-step solution

To solve the given inequality log⁡4(7)+log⁡4(2)≤log⁡4(x) \log_4(7) + \log_4(2) \le \log_4(x) , we will utilize the properties of logarithms:

  • First, apply the logarithm sum property: log⁡4(7)+log⁡4(2)=log⁡4(7×2)=log⁡4(14) \log_4(7) + \log_4(2) = \log_4(7 \times 2) = \log_4(14) .
  • Now, the inequality becomes log⁡4(14)≤log⁡4(x) \log_4(14) \le \log_4(x) .
  • Since the logarithm function is monotonically increasing when the base is greater than 1, we can simplify the inequality to 14≤x 14 \le x .

Therefore, the solution to the inequality is x≥14 x \ge 14 .

Therefore, the correct choice is 14≤x 14 \le x , which matches the given correct answer.

3

Final Answer

14≤x 14\le x

Practice Quiz

Test your knowledge with interactive questions

\( \frac{1}{\log_49}= \)

🌟 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

More Questions

Click on any question to see the complete solution with step-by-step explanations

The Sum of Logarithms

Solve the Logarithmic Inequality: ln(x+5) + ln(x) ≤ ln(4) + ln(2x)
Click to solve
Solve log₁₀3 + log₁₀4: Step-by-Step Logarithm Addition
Click to solve

Rules of Logarithms

Solve Logarithmic Inequality: log₀.₂₅7 + log₀.₂₅(1/3) < log₀.₂₅(x²)
Click to solve
Solve Base-13 Logarithmic Inequality: log₁₃(2x² + 3) ≤ log₁₃7 - log₁₃x²
Click to solve

Rules of Logarithms Combined

Solve the Logarithmic Inequality: log₁/₂(5) - log₁/₂(4) ≤ log₁/₂(x) - log₁/₂(3)
Click to solve
Solve the Logarithmic Inequality: log₂3 - log₂(x+3) ≤ 8
Click to solve