Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 7

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If $p, q \in N$ satisfy the equation $x^{\sqrt{x}} = {(\sqrt{x})}^{x}$ then $p$ & $q$ are -

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Important Questions on Logarithm and Its Applications

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 8

The expression,

\log_{p} \log_{p} \underset{\begin{matrix} n & r a d i c a l & s i g n \end{matrix}}{\underset{⏟}{\sqrt[p]{\sqrt[p]{\sqrt[p]{. . . . . . \sqrt[p]{p}}}}}}

where

p \geq 2, p \in N

, when simplified is -

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 9

Which of the following numbers are positive?

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 10

If

\log_{p} q + \log_{q} r + \log_{r} p

vanishes where

p, q

and

r

are positive reals different than unity then the value of

{(\log_{p} q)}^{3} + {(\log_{q} r)}^{3} + {(\log_{r} p)}^{3}

is -

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 11

The solution set of the system of equations,

\log_{12} x (\frac{1}{\log_{x} 2} + \log_{2} y) = \log_{2} x

and

\log_{2} x \cdot (\log_{3} (x + y)) = 3 \log_{3} x

is -

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 12

If

x_{1}

and

x_{2}

are the solution of the equation

x^{3 {\log^{3}}_{10} x - \frac{2}{3} \log_{10} x} = 100 \sqrt[3]{10}

then -

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 13

If

a^{x} = b, b^{y} = c, c^{z} = a

and

x = \log_{b} a^{2}; y = \log_{c} b^{3}

and

z = \log_{a} c^{k}

, where

a, b, c > 0

and

a, b, c \neq 1

then

k

is equal to -

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 14

If

\log_{k} x \cdot \log_{5} k = \log_{x} 5, k \neq 1, k > 0,

then

x

is equal to

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Beta Question Bank for Engineering: Mathematics>Chapter 3 - Logarithm and Its Applications>EXERCISE-2>Q 15

If

x^{\frac{3}{4} {(\log_{3} x)}^{2} + \log_{3} x - \frac{5}{4}} = \sqrt{3}

then

x

has