Properties of Logarithms

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JEE Advanced 1990

The number $lo g_{2} 7$ is

a rational number

an irrational number

JEE Advanced 1980

The least value of the expression $2 lo g_{10} x - lo g_{x} (0.01)$ , for $x > 1$ , is

- 0.01

JEE Advanced 1984

If $n$ is a natural number such that $n = p_{1}^{α_{1}} \cdot p_{2}^{α_{2}} \cdot p_{3}^{α_{3}} \dots p_{k}^{α_{k}}$ and $p_{1}, p_{2}, \dots p_{k}$ are distinct primes, then show that $ln n \geq k ln 2$ .

JEE Main 2023 (10 Apr Shift 1)

Let $a, b, c$ be the three distinct positive real numbers such that $(2 a)^{l o g_{e} a} = (b c)^{l o g_{e} b}$ and $b^{l o g_{e} 2} = a^{l o g_{e} c}$ . Then $6 a + 5 b c$ is equal to ______.

JEE Advanced 2024

Let $a = 32$ and $b = \frac{1}{5 ^{1/6} 6}$ . If $x, y \in R$ are such that $3 x + 2 y = lo g_{a} (18)^{5/4}$ and $2 x - y = lo g_{b} (1080)$ , then $4 x + 5 y$ is equal to ________.

JEE(ADVANCED)-201

The value of $((lo g_{2} 9)^{2})^{\frac{1}{l o g _{2} ( l o g _{2} 9 )}} \times (7)^{\frac{1}{l o g _{4} 7}}$ is ________.