Chapter 5

If \(x=\log _{c} b+\log _{b} c, y=\log _{a} c+\log _{c} a\) and \(z=\log _{b} a+\log _{a} b\), then find the minimum value of \(x^{2}+y^{2}+z^{2}-x y z\).

If \(\frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}\), then prove that \(a^{a} b^{b} \cdot c^{c}=1\)

Find the value of \(81^{\frac{1}{\log _{5} 3}}+27^{\log _{9} 36}+3^{\frac{4}{\log _{7} 9}}\)

Prove that \(\frac{1}{1+\log _{b} a+\log _{b} c}+\frac{1}{1+\log _{c} a+\log _{c} b}\) \(+\frac{1}{1+\log _{a} b+\log _{a} c}=1\)

Find \(x\), if \(\log _{2} x+\log _{4} x+\log _{8} x=11\)

Find \(x\), if \(\log _{2} x+\log _{4} x+\log _{8} x+\log _{16} x=\frac{25}{4}\)

If \(x=\log _{a} b c, y=\log _{b} c a\) and \(z=\log _{c} a b\), then find the value of \(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\)

If \(N=6^{\log _{10} 40} .5^{\log _{10} 36}\), then find the value of \(N+10\).

If \(x=2^{\log _{10} 3}\) and \(y=3^{\log _{i 0} 2}\), then find a relation between \(x\) and \(y\).

$$ \begin{aligned} &\text { Find the value of }\\\ &2^{\log _{40} 3-\log _{10} 5} \times 3^{\log _{40} 5-\log _{10} 2} \times 5^{\log _{10} 2-\log _{10} 3} \end{aligned} $$

If \(a=\log _{30} 3\) and \(b=\log _{30} 5\), then find the value of \(\log _{10} 8\)

If \(a=\log _{12} 18\) and \(b=\log _{24} 54\), then prove that \(a b+5(a-b)=1\)

If \(a=\log _{6} 30, b=\log _{15} 24\), then prove that \(\log _{12} 60=\left(\frac{2 a b+2 a-1}{a b+b+1}\right)\)

Find \(x\), if \(\log _{7}\left(\log _{5}(\sqrt{x+5}+\sqrt{x})\right)=0\)

\text { Solve for } x \text { and } y: 4^{\log x}=3^{\log y},(3 x)^{\log 3}=(4 y)^{\log 4}

If \(x^{18}=y^{21}=z^{28}=k\), then prove that, \(3,3 \log _{y} x, 3 \log _{z} y, 7 \log _{x} z\) are in A.P.

If \(\log _{0.3}(x-1)<\log _{0.09}(x-1)\), then find \(x\).

If \(\log _{e} \log _{5}(\sqrt{2 x-2}+3)=0\), then find the value of \(x\).

Find the least value of \(2 . \log _{10} x-\log _{x}(0.01)\) for \(x>1\).

Find \(x\), if \(4^{\log _{y} 3}+9^{\log _{2} 4}=10^{\log _{x} 83}\)

Find \(x\), if \(3^{4 \log _{9}(x+1)}=2^{2 \log _{2} x}+3\)

If \(a=\log _{24} 12, b=\log _{36} 24\) and \(c=\log _{48} 36\) then prove that \(\left(\frac{a b c+1}{b c}\right)=2\).

\(\log _{10}\left(\sin \left(x+\frac{\pi}{4}\right)\right)=\frac{1}{2}\left(\log _{10} 6-1\right)\), then find

If \(a, b, c\) are in G.P., then prove that \(\frac{1}{1+\log a}, \frac{1}{1+\log b}, \frac{1}{1+\log c}\) are in H.P.

Find \(x\), if \(5^{\log _{10} x}=50-x^{\log _{10} 5}\)

Find \(x\), if \(\log _{5}\lceil 2+\log (3+x)]=0\).

\(\log _{4}(x-1)=\log _{2}(x-3)\)

\(\log _{2} x+\log _{2}(x+3)=1 / 4\)

\(\log _{4}\left(x^{2}+x\right)-\log _{4}(x+1)=2\)

\(1+2 \log _{(x+2)} 5=\log _{5}(x+2)\)

\(\log _{2} x+\log _{4}(x+2)=2\)

\(\log _{10}(x-1)^{3}-3 \log _{10}(x-3)=\log _{10} 8\)

\(\log _{5}\left(x^{2}-3 x+3\right)>0\)

\(\log _{7}\left(\log _{5}\left(x^{2}-7 x+15\right)\right)>0\)

\(\log _{(U / 2)}\left(\log _{5}\left(x^{2}-7 x+17\right)\right)>0\)

\(\log _{(1 / 2)}\left(\log _{5}\left(\log _{2}\left(x^{2}-6 x+40\right)\right)\right)>0\)

\(\log _{3}\left(\log _{5}\left(\log _{2}\left(x^{2}-9 x+50\right)\right)\right)>0\)

\(\log _{6}\left(\frac{x-2}{6-x}\right)>0\)

\(\log _{(1 / 2)} x>\log _{(1 / 3)} x\)

0\. \(\log _{0.5}\left(x^{2}-5 x+6\right)>-1\)

\(\log _{8}\left(x^{2}-4 x+3\right)<1\)

\(\log _{(1 / 4)}\left(\frac{35-x^{2}}{x}\right) \geq-\frac{1}{2}\)

\(\log _{\left(x^{3}+6\right)}\left(x^{2}-1\right)=\log _{\left(2 x^{2}+5 x\right)}\left(x^{2}-1\right)\)

\(\log \left(3 x^{2}+x-3\right)=3 \log (3 x-2)\)

\(\log _{\left(x^{2}-1\right)}\left(x^{3}+6\right)=\log _{\left(x^{2}-1\right)}\left(2 x^{2}+5 x\right)\)

\(\log _{3}\left(x^{2}-3 x-5\right)=\log _{3}(7-2 x)\)

\(\log (\sqrt{x-1})+\frac{1}{2} \log (2 x+15)=1\)

\(\log _{(3 x+4)}\left(4 x^{2}+4 x+1\right)+\log _{(2 x+1)}\left(6 x^{2}+11 x+4\right)=4\)

\(5^{\log _{10} x}=50-x^{\log _{10} 5}\)

\(4^{\log _{9} x}-6.2^{\log _{9} x}+2^{\log _{3} 27}=0\)