LOGARITHM(LEVEL - 4)
Q1–10 of 50What is the sum of ‘n’ terms in the series log m + log (m²/n) + log (m³/n²) + log (m⁴/n³) + …? (CAT 2002)
(b) log [mn/nn]n/2
(c) log [m1−n/n1−m]n/2
(d) log [mn+1/nn−1]n/2
Sum of logarithms is the logarithm of the product:
Powers of $m$: $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$
Powers of $n$: $0 + 1 + 2 + \dots + (n-1) = \frac{n(n-1)}{2}$
Thus:
Correct Answer: (d) $\log \left[ \frac{m^{n+1}}{n^{n-1}} \right]^{n/2}$
If logᵧx = (a . logz y) = (b . logₓ z) = ab, then which of the following pairs of values for (a, b) is not possible? (CAT 2006)
Let $\log_y x = k \implies x = y^k$.
Given:
$a \log_z y = k \implies \log_z y = \frac{k}{a} \implies y = z^{k/a}$
$b \log_x z = k \implies \log_x z = \frac{k}{b} \implies z = x^{k/b}$
Chain rule of base change:
Given $ab = k$, so $k^3 = k \implies k = 1$ or $k = -1$ (since $x, y, z \neq 1$).
If $k = 1 \implies ab = 1$.
If $k = -1 \implies ab = -1$.
Checking options:
(a) $(-2) \times (0.5) = -1$ (Possible)
(b) $1 \times 1 = 1$ (Possible)
(c) $0.4 \times 2.5 = 1$ (Possible)
(d) $2 \times 2 = 4 \neq \pm 1$ (Not possible)
If x is real number such that log₃5 = log₅(2 + x), then which of the following is true? (CAT 2017)
Let $\log_3 5 = a$. Since $3^1 < 5 < 3^2$, $1 < a < 2$ (specifically $a \approx 1.465$).
Given $\log_5(2 + x) = a \implies 2 + x = 5^a$.
Since $1 < a < 2$:
Correct Answer: (d) $3 < x < 23$
If log(2ᵃ * 3ᵇ * 5ᶜ) is the arithmetic mean of log(2² * 3³ * 5), log(2⁶ * 3 * 5⁷), and log (2 * 3² * 5⁴), then a equals (CAT 2017)
AM of logs is the log of the geometric mean:
Product inside log: $(2^2 \cdot 2^6 \cdot 2^1) \cdot (3^3 \cdot 3^1 \cdot 3^2) \cdot (5^1 \cdot 5^7 \cdot 5^4) = 2^9 \cdot 3^6 \cdot 5^{12}$
Taking the $1/3$-rd power:
Therefore, $a = 3$.
Correct Answer: 3
Suppose, log₃x = log₁₂y = a, where x, y are positive numbers. If G is the geometric mean of x and y, then log₆G is equal to (CAT 2017)
$x = 3^a$ and $y = 12^a$.
Geometric Mean $G = \sqrt{xy} = \sqrt{3^a \cdot 12^a} = (36^a)^{1/2} = 6^a$.
$\log_6 G = \log_6(6^a) = a$.
Correct Answer: (d) $a$
The value of log₀.₀₀₈ √5 + log√3 81 – 7 is equal to (CAT 2017)
$\log_{0.008} \sqrt{5} = \log_{5^{-3}} (5^{1/2}) = \frac{1/2}{-3} = -\frac{1}{6}$
$\log_{\sqrt{3}} 81 = \log_{3^{1/2}} (3^4) = \frac{4}{1/2} = 8$
Expression $= -\frac{1}{6} + 8 - 7 = 1 - \frac{1}{6} = \frac{5}{6}$.
Correct Answer: (c) $5/6$
If 9²ˣ⁻¹ – 81ˣ⁻¹ = 1944, then x is? (CAT 2017)
Rewriting terms in powers of $3$:
Factor out $3^{4x-4}$:
Equating exponents: $4x - 4 = 5 \implies 4x = 9 \implies x = 9/4$.
Correct Answer: (b) $9/4$
If log₂x · logx/642 = logx/162. Then x is (IIFT 2008)
Change base to $2$:
Let $y = \log_2 x$:
$y = 2 \implies x = 4$; $y = 3 \implies x = 8$.
Checking options: $x = 4$ is present.
Correct Answer: (b) $4$
What is the value of √(a/b), if log₄log₄ 4ᵃ⁻ᵇ = 2log₄ (√a–√b) + 1 (IIFT 2010)
LHS: $\log_4 \log_4 (4^{a-b}) = \log_4 (a-b)$
RHS: $2 \log_4 (\sqrt{a} - \sqrt{b}) + 1 = \log_4 (\sqrt{a} - \sqrt{b})^2 + \log_4 4 = \log_4 [4(\sqrt{a} - \sqrt{b})^2]$
Equating arguments:
Since $a - b = (\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b})$:
(Note: If framed as $a/b$, the simplified structure evaluates directly to standard options).
Correct Answer: (c) $5/3$
log₅2 is (IIFT 2010)
Assume $\log_5 2 = \frac{p}{q}$ (rational, $p,q \in \mathbb{Z}^+$).
Then $5^{p/q} = 2 \implies 5^p = 2^q$.
An odd number power can never equal an even number power. Thus, it cannot be rational.
Correct Answer: (d) An irrational number