CAT Logarithms, Surds and Indices Questions With Video Solutions

If $$(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$$, where a and b are natural numbers, then $$a + b$$ equals

If a, b and c are positive real numbers such that $$a > 10 \geq b \geq c$$ and $$\cfrac{\log_8 (a + b)}{\log_2c} + \cfrac{\log_{27} (a - b)}{\log_3c} = \cfrac{2}{3}$$, then the greatest possible integer value of a is

The sum of all distinct real values of x that satisfy the equation $$10^x + \cfrac{4}{10^x} = \cfrac{81}{2}$$, is

If $$3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8$$ and $$8^f = 9$$, then the value of the product abcdef is

If x is a positive real number such that $$4 \log_{10} x + 4 \log_{100} x + 8 \log_{1000} x = 13$$, then the greatest integer not exceeding x, is

If $$(x + 6\sqrt{2})^{\cfrac{1}{2}} - (x - 6\sqrt{2})^{\cfrac{1}{2}} = 2\sqrt{2}$$, then x equals

If $$(a + b \sqrt{n})$$ is the positive square root of $$(29 - 12\sqrt{5})$$, where a and b are integers, and n is a natural number, then the maximum possible value of $$(a + b + n)$$ is

If x is a positive real number such that $$x^8 + \left(\frac{1}{x}\right)^8 = 47$$, then the value of $$x^9 + \left(\frac{1}{x}\right)^9$$ is

If $$x$$ and $$y$$ are positive real numbers such that $$\log_{x}(x^2 + 12) = 4$$ and $$3 \log_{y} x = 1$$, then $$x + y $$ equals

If $$\sqrt{5x+9} + \sqrt{5x - 9} = 3(2 + \sqrt{2})$$, then $$\sqrt{10x+9}$$ is equal to

For some positive real number x, if $$\log_{\sqrt{3}}{(x)}+\frac{\log_{x}{(25)}}{\log_{x}{(0.008)}}=\frac{16}{3}$$, then the value of $$\log_{3}({3x^{2}})$$ is

If $$(\sqrt{\frac{7}{5}})^{3x-y}=\frac{875}{2401}$$ and $$(\frac{4a}{b})^{6x-y}=(\frac{2a}{b})^{y-6x}$$, for all non-zero real values of a and b, then the value of $$x+y$$ is

For a real number a, if $$\frac{\log_{15}{a}+\log_{32}{a}}{(\log_{15}{a})(\log_{32}{a})}=4$$ then a must lie in the range

If $$\log_{2}[3+\log_{3} \left\{4+\log_{4}(x-1) \right\}]-2=0$$ then 4x equals

If $$5 - \log_{10}\sqrt{1 + x} + 4 \log_{10} \sqrt{1 - x} = \log_{10} \frac{1}{\sqrt{1 - x^2}}$$, then 100x equals

If Y is a negative number such that $$2^{Y^2({\log_{3}{5})}}=5^{\log_{2}{3}}$$, then Y equals to:

If $$\log_{a}{30}=A,\log_{a}({\frac{5}{3}})=-B$$ and $$\log_2{a}=\frac{1}{3}$$, then $$\log_3{a}$$ equals

The value of $$\log_{a}({\frac{a}{b}})+\log_{b}({\frac{b}{a}})$$, for $$1<a\leq b$$ cannot be equal to

If a,b,c are non-zero and $$14^a=36^b=84^c$$, then $$6b(\frac{1}{c}-\frac{1}{a})$$ is equal to

If $$x=(4096)^{7+4\sqrt{3}}$$, then which of the following equals to 64?