Logarithms — CAT Previous-Year Questions

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 postive 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

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

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