HARD

Earn 100

For the equation $1 + {l o g}_{x} (\frac{4 - x}{10}) = ({l o g}_{10} ({l o g}_{10} p) - 1) {l o g}_{x} 10$ , match the value of p

given in column-II corresponding to the number of solutions given in column-I.

Column-I Column-II

$(A)$ for no solution $(P)$ undefined

$(B)$ for exactly one solution $(Q)$ undefined

$(C)$ for two distinct solutions $(R)$ undefined

$(S)$ undefined , ∞)

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Important Questions on Logarithm and its Applications

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

\log_{8} x + \log_{4} x + \log_{2} x = 11, x =

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Let

x, y

be real numbers such that

x > 2 y > 0

and

2 \log (x - 2 y) = \log x + \log y

. Then the possible value(s) of

\frac{x}{y}

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

\log_{e} (y^{2}) = \tan (\frac{1}{x}),

then

y =

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

For

0 < m < 1

, which one of the following is correct?

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Let

n

be a positive integer such that

\log_{2} \log_{2} \log_{2} \log_{2} \log_{2} (n) < 0 < \log_{2} \log_{2} \log_{2} \log_{2} (n)

. Let

l

be the number of digits in the binary expansion of

n

. Then the minimum and the maximum possible values of

l

are

HARD

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Let

S

be the sum of the digits of the number

15^{2} \times 5^{18}

in base

10

. Then,

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Let

n

be the smallest positive integer such that

1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} \geq 4

. Which one of the following statements is true?

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

f (x) = \log_{e} (\frac{1 - x}{1 + x}), |x| < 1,

then

f (\frac{2 x}{1 + x^{2}})

is equal to

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

x + \log_{10} (1 + 2^{x}) = x \log_{10} 5 + \log_{10} 6,

then the value of

x

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

2 \log (x + 1) - \log (x^{2} - 1) = \log 2,

then

x =

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Let a complex number

z, |z| \neq 1

, satisfy

\log_{\frac{1}{\sqrt{2}}} (\frac{|z| + 11}{{(|z| - 1)}^{2}}) \leq 2

. Then, the largest value of

|z|

is equal to _________.

HARD

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Let

f

be a function defined on the set of all positive integers such that

f (x y) = f (x) + f (y)

for all positive integers

x, y .

f (12) = 24

and

f (8) = 15

, the value of

f (48)

HARD

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

If for $x \in (0, \frac{π}{2}), \log_{10} \sin x + \log_{10} \cos x = - 1$ and $\log_{10} (\sin x + \cos x) = \frac{1}{2} (\log_{10} n - 1), n > 0$ , then the value of $n$ is equal to :

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

If $\log 3^{x + 4} = \log 729$ then the value of $x$ will be

HARD

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

The solution set of the inequality

\log_{\sin (π / 3)} (x^{2} - 3 x + 2) \geq 2

HARD

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

\log_{(3 x - 1)} (x - 2) = \log_{(9 x^{2} - 6 x + 1)} (2 x^{2} - 10 x - 2),

then

x

equals-

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

The number of solution pairs

(x, y)

of the simultaneous equations

\log_{1 / 3} (x + y) + \log_{3} (x - y) = 2

and

2 y^{2} = 512^{x + 1}

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Which of the following is negative ?

MEDIUM

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Set of all the values of

x

satisfying the inequality

\log_{x + \frac{1}{x}} (\log_{2} \frac{x - 1}{x + 2}) > 0

EASY

Mathematics>Algebra>Logarithm and its Applications>Logarithmic Equations

Solve

\log_{3} (5 x - 1) > 0

Column-I	Column-II
$(A)$ for no solution	$(P)$ undefined
$(B)$ for exactly one solution	$(Q)$ undefined
$(C)$ for two distinct solutions	$(R)$ undefined
	$(S)$ undefined , ∞)

Important Questions on Logarithm and its Applications

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