CAT Logarithms & Indices questions (2017–2025) — actual Quant questions with solutions

The sum of all possible real values of x for which log_x-3(x² − 9) = log_x-3(x + 1) + 2 is :

X Maro G Strategy | Surds and Indices | Moderate

If 12¹²^x × 4²⁴^{x +}¹² × 5²^y = 8⁴^z × 20¹²^x × 243^3x-6 , where x, y and z are natural numbers, then x+y+z equals_____

The sum of digits of the number (625)⁶⁵ × (128)³⁶ is ______

If log₆₄(x²) + log₈(√y) + 3 log₅₁₂(√(y)z) = 4, where x, y and z are positive real numbers, then the minimum possible value of (x + y + z) is

Concepts of Indices | HARD | CAT 2025 Slot 1

For any natural number k, let aₖ = 3ᵏ. The smallest natural number m for which (a₁)¹ × (a₂)² × ... × (a₂₀)²⁰ < (a₂₁ × a₂₂ × ... × a₂₀₊ₘ)⁵ is:

The number of distinct integers n for which log_1/4 of (n² − 7n + 11) > 0 is _____.

Indices + Logs

If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8, and 8^f = 9, then what is the value of abcdef?

\( a \) and \( b \) are natural numbers, then the value of \( a + b \) is: \[ (a + b\sqrt{3})^2 = 52 + 30\sqrt{3} \]

Logs + Maxima

Logarithm Expression Problem If \( a, b, c \) are positive real numbers such that: \[ a > 10 \geq b \geq c \quad \text{and} \quad \frac{\log_8(a + b)}{\log_2 c} + \frac{\log_{27}(a - b)}{\log_3 c} = \frac{2}{3} \] Then the greatest possible integer value of \( a \) is?

Logs DPAC Shortcut

Logarithm Equation Question 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 > 10 ≥ b ≥ c and given a logarithmic equation involving a, b, c, find the greatest possible integer value of a.

If 3^a = 4, 4^b = 5, 5^c = 6, 6^d = 7, 7^e = 8, 8^f = 9, then find the value of a×b×c×d×e×f.

Solve: 10^x + 4/(10^x) = 81/2. Find sum of all real values of x.

The sum of all real values of k for which the equation x²²⁷⁶ = x²²⁷⁶ holds true is. If k = 1, 32768, then the value of k is

If x is a positive real number such that 4 log₁₀ x + 4 log₁₀₀ x + 8 log₁₀₀₀ x = 13, then the greatest integer not exceeding x is

Let \( n \) be any natural number such that \( 5^{n-1} < 3^{n+1} \). The least integer value of \( m \) that satisfies \( 3^{n+1} < 2^{n+m} \) for all such \( n \) is?

The number of distinct integer values of n satisfying 4−log²n/3−log4ⁿ<0, is

For a real number a, if log15^a+log32^a/(log15a)(log32a)= = 4 then a must lie in the range

If log2^[3 + log3^{4 + log4^{x − 1}}] − 2 = 0 then 4x equals

If x¹ = −1 and x^m = x^m + 1 + (m + 1) for every positive integer m, then x¹⁰⁰ equals

Let loga³⁰ = A, loga⁵³ = −B and log2^a = 1/3, then log3^a equals

2×4×8×16/(log2⁴)²(log4⁸)³(log8¹⁶)⁴ equals

If a,b,c are non-zero and 14^a = 36^b = 84^c, then 6b(1/c − 1/a) is equal to

If log4⁵ = (log4^y) (log6^ √5), then y equals

The number of real-valued solutions of the equation 2^x + 2^−x = 2 − (x − 2)² is

If y is a negative number such that 2^y2log3⁵ = 5^log2³, then y equals

If 5^x – 3^y = 13438 and 5^x-1 + 3^y+1 = 9686 , then x+y equals

If (5.55)^x = (0.555)^y = 1000, then the value of 1/x − 1/y is

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 p³ = q⁴ = r⁵ = s⁶, then the value of logs^ (pqr) is equal to

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