Examples with solutions for Rules of Logarithms Combined: Inequality

Exercise #1

Is inequality true?

\log_{\frac{1}{4}}9<\frac{\log_57}{\log_5\frac{1}{4}}

Video Solution

Answer

Yes, since:

\log_{\frac{1}{4}}9<\log_{\frac{1}{4}}7

Exercise #2

log23log2(x+3)8 \log_23-\log_2(x+3)\le8

Video Solution

Answer

x32563 x\ge\frac{3}{256}-3

Exercise #3

log35x×log179log174 \log_35x\times\log_{\frac{1}{7}}9\ge\log_{\frac{1}{7}}4

Video Solution

Answer

0 < x\le\frac{1}{245}

Exercise #4

\log_{\frac{1}{3}}e^2\ln x<3\log_{\frac{1}{3}}2

Video Solution

Answer

\sqrt{8} < x

Exercise #5

x=? x=\text{?}

ln(x+5)+lnxln4+ln2x \ln(x+5)+\ln x≤\ln4+\ln2x

Video Solution

Answer

0 < X \le 3

Exercise #6

x=? x=\text{?}

log125log124log12xlog123 \log_{\frac{1}{2}}5-\log_{\frac{1}{2}}4\le\log_{\frac{1}{2}}x-\log_{\frac{1}{2}}3

Video Solution

Answer

0 < x\le3.75

Exercise #7

What is the domain of X so that the following is satisfied:

\frac{\log_{\frac{1}{8}}2x}{\log_{\frac{1}{8}}4}<\log_4(5x-2)

Video Solution

Answer

\frac{2}{3} < x

Exercise #8

Find the domain X where the inequality exists

2\log_3x<\log_3(x^2+2x-12)

Video Solution

Answer

6 < x

Exercise #9

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

x=? x=\text{?}

Video Solution

Answer

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

Exercise #10

Find the domain of X given the following:

\log_{\frac{1}{7}}(x^2+3x)<2\log_{\frac{1}{7}}(3x+1)

Video Solution

Answer

No solution

Exercise #11

Given 0<X , find X

log4x×log564log5(x3+x2+x+1) \log_4x\times\log_564\ge\log_5(x^3+x^2+x+1)

Video Solution

Answer

No solution

Exercise #12

Given X>1 find the domain X where it is satisfied:

log3(x2+5x+4)log3x<logx12 \frac{\log_3(x^2+5x+4)}{\log_3x}<\log_x12

Video Solution

Answer

1 < x < -2.5+\frac{\sqrt{57}}{2}

Exercise #13

x=? x=\text{?}

log13(2x2+3)log132log137log13x2 \log_{13}(2x^2+3)-\log_{13}2\le\log_{13}7-\log_{13}x^2

Video Solution

Answer

2x2 -\sqrt{2}\le x\le\sqrt{2}