Logarithms CAT PYQs , 29 Questions with Solutions

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

$\log_{x-3}(x^2-9) = \log_{x-3}(x+1) + 2$

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 $a, b$ and $c$ are positive real numbers such that $a>10 \geq b \geq c$ 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 $\log_{x}\left(x^2+12\right)=4$ and $3 \log_{y} x=1$, then $x+y$ equals

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 $\log_3\left(3 x^2\right)$ is

For a real number $x$, if $\frac{1}{2}, \frac{\log_{3}(2^x - 9)}{\log_{3} 4}, \frac{\log_{5}\left(2^x + \frac{17}{2}\right)}{\log_{5} 4}$ are in an arithmetic progression, then what is the common difference?

The number of distinct integer values of $n$ satisfying $\frac{4-\log _2 n}{3-\log _4 n}\lt0$ , is

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 $\log_{2}\!\big[\,3 + \log_{3}\{\,4 + \log_{4}(x-1)\,\}\big] - 2 = 0$ then $4x$ equals

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 $y$ is a negative number such that $2^{y^2 \log_3 5} = 5^{\log_2 3}$, then $y$ equals

If $\log_{4} 5 = (\log_{4} y) (\log_{6} \sqrt{5})$, then $y$ equals

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

$\frac{2×4×8×16}{(log_{2}4)^{2}(log_{4}8)^{3}(log_{8}16)^{4}}$ equals

Let $\log_a 30 = A$, $\log_a \frac{5}{3} = -B$ and $\log_2 a = \frac{1}{3}$, then $\log_3 a$ equals

If $(5.55)^x = (0.555)^y = 1000$, then the value of $\frac{1}{x} - \frac{1}{y}$ is

Let $x$ and $y$ be positive real numbers such that $\log_{5} (x + y) + \log_{5} (x - y) = 3$, and $\log_{2} y - \log_{2} x = 1 - \log_{2} 3$. Then $xy$ equals

If x is a real number ,then $\sqrt{log_{e}\frac{4x - x^2}{3}}$ is a real number if and only if

If $\log_{12} 81 = p$, then $3 \left( \frac{4 - p}{4 + p} \right)$ is equal to:

If log 2 (5 + log 3 a) = 3 and log 5 (4a + 12 + log 2 b) = 3, then a + b is equal to

If $p^3 = q^4 = r^5 = s^6$, then the value of $\log_s (pqr)$ is equal to

$\frac{1}{\log_2 100}$ - $\frac{1}{\log_4 100}$ + $\frac{1}{\log_5 100}$ - $\frac{1}{\log_{10} 100}$ + $\frac{1}{\log_{20} 100}$ - $\frac{1}{\log_{25} 100}$ + $\frac{1}{\log_{50} 100}$ = ?

The value of $\log_{0.008} \sqrt{5} + \log_{\sqrt{3}} 81 - 7$ is equal to :

Suppose, $\log_{3} x = \log_{12} y = a$, where $x, y$ are positive numbers. If $G$ is the geometric mean of $x$ and $y$, and $\log_{6} G$ is equal to:

If $x$ is a real number such that $\log_{3} 5 = \log_{5} (2 + x)$, then which of the following is true?

If $\log(2^a \times 3^b \times 5^c)$ is the arithmetic mean of $\log(2^2 \times 3^3 \times 5)$, $\log(2^6 \times 3 \times 5^7)$, and $\log(2 \times 3^2 \times 5^4)$, then $a$ equals (TITA).