Logarithms — CAT Previous-Year Questions

The number of distinct integers $n$ for which $\log _{\left(\frac{1}{4}\right)}\left(n^2-7 n+11\right)>0$, is

If $\log _ { 64 } x ^ { 2 } + \log _ { 8 } \sqrt { y } + 3 \log _ { 512 } ( \sqrt { y } z ) = 4$, where $x , y$ and $z$ are positive real numbers, then the minimum possible value of $( x + y + z )$ is

The sum of all possible real values of $x$ for which $\log _ { x - 3 } \left( x ^ { 2 } - 9 \right) = \log _ { x - 3 } ( x + 1 ) + 2$, is

If x is a positive number such that 4 log₁₀x + 4 log₁₀₀x + 8 log₁₀₀₀x = 13, then the greatest integer not exceeding x, is

If a, b and c are positive real numbers such that a > 10 ≥ b ≥ c and $\frac{{\log }_8(a+b)}{{\log }_{2}c}$ + $\frac{{\log }_{27}(a-b)}{{\log }_{3}c}$ = $\frac{2}{3}$, then the greatest possible integer value of a is

If x and y are positive real numbers such that log_x (x² + 12) = 4 and 3log_y x = 1, then x + y equals?

For some positive ral number x, if ${\log }_{\sqrt{3}} \left(x\right)$ + $\frac{{\log }_x 25}{{\log }_x (0.008)}$ = $\frac{16}{3}$, then the value of ${\log }_3 \left(3x^2\right)$ is

For a real number x, if $\frac{1}{2}$, $\frac{{\log }_3(2^x-9)}{{\log }_{3}4}$, and $\frac{{\log }_5(2^x+{\displaystyle \frac{17}{2}})}{{\log }_{5}4}$ are in arithmetic progression, then the common difference is

The number of distinct integer values of n satisfying $\frac{4-{\log }_2\mathrm{n}}{3-{\log }_4\mathrm{n}}$ < 0, is

If 5 – ${\log }_{10}\left(\sqrt{1+x}\right)$ + 4${\log }_{10}\left(\sqrt{1-x}\right)$ = ${\log }_{10}\left(\frac{1}{\sqrt{1-x^2}}\right)$, then 100x equals

log₂ [3 + log₃ {4 + log₄ (x - 1)}] - 2 = 0, then 4x equals

For a real number a, if $\frac{lo{g}_{15}a+lo{g}_{32}a}{lo{g}_{15}a\times lo{g}_{32}a}$ = 4, then a must lie in the range

If y is a negative number such that $2^{{\mathrm{y}}^2{\log }_{3}5}=5^{{\log }_{2}3}$, then y equals

If log₄ 5 = (log₄ y)(log₆ √5), then y equals

    The value of ${\log }_a\left(\frac{a}{b}\right)+{\log }_b\left(\frac{b}{a}\right)$, for 1 < a ≤ b cannot be equal to

$\frac{2\times 4\times 8\times 16}{\left({{\log }_{2}4)}^2\right({{\log }_{4}8)}^3({{\log }_{8}16)}^4}$ equals

If log_a30 = A, log_a(5/3) = -B and log₂a = 1/3, then log₃a equals

Let x and y be positive real numbers such that log₅(x + y) + log₅(x - y) = 3, and log₂y - log₂x = 1 - log₂3. Then xy equals

If x is a real number, then $\sqrt{{\log }_e\left(\frac{4x-x^2}{3}\right)}$ is a real number if and only if

If x is a positive quantity such that 2^x = $3^{{\log }_5\left(2\right)}$ , then x is equal to

If log₁₂81 = p, then $3\left(\frac{4-p}{4+p}\right)$ is equal to

If log₂(5 + log₃ a) = 3 and log₅(4a + 12 + log₂ b) = 3, then a + b is equal to

If p^₃ = q⁴ = r⁵ = s⁶, then the value of log_s(pqr) is equal to

The smallest integer n for which 4ⁿ > 17¹⁹ holds, is closest to

$\frac{1}{l{\mathrm{og}}_{2}100}-\frac{1}{{\log }_{4}100}+\frac{1}{{\log }_{5}100}-\frac{1}{{\log }_{10}100}+\frac{1}{{\log }_{20}100}-\frac{1}{{\log }_{25}100}+\frac{1}{{\log }_{50}100}?$

The value of log_0.008√5 + log_√381 – 7 is equal to:

If x is a real number such that log₃5 = log₅(2 + x), then which of the following is true?

If log_yx = a × log_zy = b × log_xz = ab, then which of the following pairs of values for (a, b) is not possible?

If x ≥ y and y > 1, then the value of the expression

${\log }_x\left(\frac{x}{y}\right)+{\log }_y\left(\frac{y}{x}\right)$ can never be

Let u = (log₂ x)² – 6 log₂ x + 12 where x is a real number. Then the equation x^u = 256, has

If $\frac{1}{3}{\log }_{3}M+3{\log }_{3}N=1+{\log }_{0.008}5,$ then

If log₁₀ x - log₁₀ $\sqrt{x}$ = 2 log_x 10, then a possible value of x is given by:

What is the sum of n terms in the series

$\log m+\log \frac{m^2}{n}+\log \frac{m^3}{n^2}+\log \frac{m^4}{n^3}+\cdots +\log \frac{m^n}{n^{n-1}}?$

Directions: Each question is followed by two statements I and II. Mark:
1. if the question can be answered by any one of the statements alone, but cannot be answered by using the other statement alone.
2. if the question can be answered by using either statement alone.
3. if the question can be answered by using both the statements together, but cannot be answered by using either statement alone.
4. if the question cannot be answered even by using both the statements together.

What is the distance x between two cities A and B in integral number of kilometres?
I. x satisfies the equation ${\log }_{2}x=\sqrt{x}$
II. x ≤ 10 km

If log₂ [log₇ (x² - x + 37)] = 1, then what could be the value of ‘x’?

If log₇ log₅ $\left(\sqrt{x+5}+\sqrt{x}\right)$ = 0, find the value of x.

log₆ 216$\sqrt{6}$ is