Chapter 5

Solve for \(x: x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6\)

If \(a, b, c\) are in G.P. then \(\log _{2016} a, \log _{2016} b, \log _{2016} c\) are in (a) G.P. (b) A.P. (c) H.P. (d) A.G.P.

Find the value of (i) \(\log _{2} 32\) (ii) \(\log _{3}\left(\frac{1}{243}\right)\) (iii) \(\log _{5 \sqrt{5}} 5\)(iv) \(\log _{100}(0.1)\) (v) \(10^{\log _{10} m+\log _{10} n}\) (vi) \(\log _{3} \log _{5} \log _{3}(243)\)

Solve for \(x: \log \left|\frac{x^{2}-x-1}{x^{2}+x-2}\right|=0\)

Find the value of (i) \(\log _{2} \log _{3} \log _{4}(64)\) (ii) \(3^{-\frac{1}{3} \log _{5} 7}\) (iii) \(2^{2-\log _{2} 5}\) (iv) \(2^{\log _{3} 5}-5^{\log _{3} 2}\) (v) \(\log _{9} 27-\log _{27} 9\)

Solve for \(x:\left|4+\log _{1 / 7} x\right|=2+\left|2+\log _{1 / 7} x\right|\)

If \(\frac{1}{\log _{a} x}+\frac{1}{\log _{c} x}=\frac{2}{\log _{b} x}\), then \(a, b, c\) are in (a) A.P. (b) G.P. (c) H.P. (d) A.G.P.

Find the value of (i) \(\log _{10} \tan 40^{\circ}+\log _{10} \tan 41^{\circ}+\log _{10} \tan 42^{\circ}\) \(+\ldots \ldots+\log _{10} \tan 50^{\circ}\) (ii) \(\log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ}+\log _{10} \tan 3^{\circ}\) \(+\ldots \ldots \ldots . .+\log _{10} \tan 89^{\circ}\) (iii) \(\log _{3} 4 \cdot \log _{4} 5 \cdot \log _{5} 6 \cdot \log _{6} 7 \cdot \log _{7} 8 \cdot \log _{8} 9\)

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

If \(x=\log _{3} 5\) and \(y=\log _{27} 25\), then (a) \(x>y\) (b) \(x=y\) (c) \(x

Find the value of (i) \(x\), if \(\log _{5} a \cdot \log _{a} x=2\), (ii) \(x\), if \(\log _{k} x \cdot \log _{5} k=\log _{k} 5\), where \(k \neq 1, k>0\) (iii) \(A+B+10\), if \(A=\log _{2} \log _{2} \log _{4} 256\) and \(B=2 \log _{\sqrt{2}} 2\), (iv) \(\log _{\sqrt{3}} 300 .\), if \(a=\log _{\sqrt{3}} 5, b=\log _{\sqrt{3}} 2\)

Solve the system of equations: \(\left\\{\begin{array}{c}\log _{y} x-\log _{x} y=\frac{8}{3} \\ x y=16\end{array}\right.\)

If \(\log _{10} 2, \log _{10}\left(2^{x}+1\right), \log _{10}\left(2^{x}+3\right)\) are in A.P. then (a) \(x=0\) (b) \(x=1\) (c) \(x=\log _{10} 2\) (d) \(x=\frac{1}{2} \log _{2} 5\)

Solve for \(x\) : \(\log \left(3 x^{2}+12 x+19\right)-\log (3 x+4)+\log _{32} 4\) \(=1-\log _{1 / 6}(\sqrt[5]{256})\)

If \(\log _{a}(a b)=x\), then \(\log _{b}(a b)\) is (a) \(\frac{x}{1-x}\) (b) \(\frac{x}{1+x}\) (c) \(\frac{x}{x-2}\) (d) None

Solve for \(x\) : \(\log ^{2}(4-x)+\log (4-x) \cdot \log \left(x+\frac{1}{2}\right)-2 \log ^{2}\left(x+\frac{1}{2}\right)=0\)

The value of \(4^{2 \log _{9} 3}\) is (a) \(\overline{9}\) (b) 2 (c) 4 (d) 3

If \(n=1983\), then prove that \(\frac{1}{\log _{2} n}+\frac{1}{\log _{3} n}+\frac{1}{\log _{4} n}+\ldots . .+\frac{1}{\log _{1983} n}=1\)

Solve for \(x\) : \(\log _{3 / 4}\left(\log _{8}\left(x^{2}+7\right)\right)+\log _{1 / 2}\left(\log _{1 / 4}\left(x^{2}+7\right)^{-1}\right)=-2\)

If \(\log _{7} 2=x\), then \(\log _{49}(28)\) is (a) \(\left(x+\frac{1}{2}\right)\) (b) \(\left(x-\frac{1}{2}\right)\) (c) \(-\left(x-\frac{1}{2}\right)\) (d) \(-\left(x+\frac{1}{2}\right)\)

Determine \(b\) satisfying (i) \(\log _{\sqrt{8}} b=3 \frac{1}{3}\) (ii) \(\log _{\sqrt{8}} b=3^{\frac{1}{3}}\) (iii) \(\log _{a} 2 \cdot \log _{b} 625=\log _{10} 16 \cdot \log _{a} 10\) (iv) \(\log _{3} \log _{2} \log _{\sqrt{5}}(625)=b\)

Solve for \(x\) : \(\log _{10}\left(x^{2}-x-6\right)-x=\log _{10}(x+2)-4\)

If \(\log _{2016}\left(\log _{5}(\sqrt{2 x-2}+3)\right)=0\), then \(x\) is (a) \(1 / 3\) (b) \(1 / 2\) (c) 3 (d) 2

If \(\log _{a} a b=x\), then find the value of \(\log _{b} a b\).

Solve for \(x\) : \(\frac{1}{2} \log _{5}(x+5)+\log _{5}(\sqrt{x-3})=\frac{1}{2} \log _{5}(2 x+1)\)

If \(\frac{1}{\log _{2} \pi}+\frac{1}{\log _{6} \pi}>x\), then \(x\) is (a) 2 (b) 3 (c) 4 (d) 5

If \(\log _{10} 2=x\), then find the value of \(\log _{10} 5\).

Solve for \(x\) : \(\frac{3}{2} \log _{4}(x+2)^{2}+3=\log _{4}(4-x)^{3}+\log _{4}(6+x)^{3}\)

The value of \(5^{\log _{2} 7}-7^{\log _{2} 5}\) is (a) 5 (b) 0 (c) 7 (d) 2

Solve for \(x\) : \(\frac{1+\log _{2}(x-4)}{2 \log _{2}(\sqrt{x+3}-\sqrt{x-3})}=1\)

If \(\log _{10} 2=x\), then \(\log _{10} 5\) is (a) 1 (b) \(1-x\) (c) \(x+1\) (d) \(2 x\)

Find the value of \(\log _{12} 54\), where \(b=\log _{12} 24\).

Solve for \(x\) : \(\left(1+\frac{1}{2 x}\right) \log 3=\log \left(\frac{\sqrt[x]{3}+27}{4}\right)\)

The number of real solutions of \(\log _{2} x+\log _{4}(x+2)=2\) is (a) 1 (b) 2 (c) 3 (d) 0

Find the value of \(\frac{1}{\log _{2} 36}+\frac{1}{\log _{3} 36}\)

Solve for \(x\) : \(4^{\log _{10} x+1}-6^{\log _{10} x}-2 \cdot 3^{\log _{19} x^{2}+2}=0\)

Prove that \(\frac{1}{\log _{3} \pi}+\frac{1}{\log _{4} \pi}>2\)

Solve for \(x\) : \(\log _{3}(\sqrt{x}+|\sqrt{x}-1|)^{2}=\log _{3}(4 \sqrt{3}-3+4|\sqrt{x}-1|)\)

The number of real solutions of \(x^{\log _{\sqrt{x}}(x-2)}=9\) is (a) 4 (b) 3 (c) 2 (d) 1

Simplify: \(7 . \log \frac{16}{15}+5 \cdot \log \frac{25}{24}+3 \cdot \log \frac{81}{80}\)

The value of \(\left(\frac{1}{\log _{3} \pi}+\frac{1}{\log _{4} \pi}\right)\) lies between (a) \((1,2)\) (b) \((2,3)\) (c) \((3,4)\) (d) \((0,1)\)

If \(a^{2}+b^{2}=7 a b\), then prove that \(\log \frac{1}{3}(a+b)=\frac{1}{2}(\log a+\log b)\)

The number of real roots of \(x \ln x-1=0\) is (a) 2 (b) 1 (c) 3 (d) infinite

The number of real roots of \(2-x-\ln x=0\) is (a) 1 (b) 2 (c) 0 (d) infinite

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

If \(3^{x}=10-\log _{2} x\), then \(x\) is (a) 0 (b) 1 (c) 2 (d) 3

If \(\left|1-\log _{1 / 5} x\right|+2=\left|3-\log _{1 / 5} x\right|\), then \(x\) is (a) 2 (b) 5 (c) 1 (d) 3

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

If \(\log _{3} 2, \log _{3}\left(2^{x}-5\right), \log _{3}\left(2^{x}-\frac{7}{2}\right)\) are in A.P., then find the value of \(x\)

If \(y=a^{\overline{1-\log _{a} x}}, z=a^{\overline{1-\log a y}}\), then prove that, \(x=a^{\frac{1}{1-\log _{a} z}}\).